Kinematics - City Tech OpenLab

A. Maller

KinematicsChapter 2

A. MallerHow fast is the imperial cruiser traveling?

A. Maller

It takes about 20s for the imperial cruiser to enter the screen.

You can find online that the length of the cruiser is 1600m.

So the speed of the cruiser is 1600m/20s = 80m/s.

80m/s × 1km/1000m × 3600s/1hr = 288 km/hr

179 mph

A. Maller

Motion

Kinematics is the study of motion.

To start out easy we will just consider one object moving in one dimension. We will add a second dimension soon.

So basically our object starts at some place x1 and ends up at a second place x2 after some time t.

We can then define an average velocity or speed that the object had during that time.

v =x2 � x1

A Δ means the change in some quantity

A bar over a letter means the average value of that quantity

A. Maller

Frames of ReferenceFirst let us discuss what x1 and x2 mean. Say we have a bowling ball that moves from one location to another.

We can place a coordinate axis behind them giving them x and y positions. Note we can place the coordinate axis anywhere we want, it only exists in our imagination.

It makes sense to choose a place for the axis that makes our life easiest. For example, having the motion along the axis and the original position be at x=0.

A. Maller

Instantaneous VelocityIf you take a shorter and shorter time interval over which to ask what the average velocity is eventually you get to the instantaneous velocity or what we just call velocity.

Notice that earlier when average velocity was introduced it was displacement dived by time, t. However, it really should be the change in time, Δt. If someone left at 2PM and arrived at 4:30PM the change in time would be 2.5 hours, that would be the time we want. However, we would also say the time the trip took was 2.5 hours. Time is a little tricky because we often don’t distinguish between the time something occurred at and the change in time between two events.

Sometimes we run into the same problems with distance. We are always only interested in the change for either.

v = lim�t!0

A. Maller

Example 3.3The position of a particle is given by x(t)=3.0t+0.5 t3 m. a) Find the instantaneous velocity at t=2.0 b) Calculate the average velocity between 1.0 s and 3.0 s.

v(t) =dx

dt(3.0t+ 0.5t3) = 3.0 + 1.5t2m/s

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v(2s) = 3.0 + 1.5(2)2 = 9.0m/s<latexit sha1_base64="0jdtyYEV8iYGTFB7Kc9nvNO4968=">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</latexit><latexit sha1_base64="0jdtyYEV8iYGTFB7Kc9nvNO4968=">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</latexit><latexit sha1_base64="0jdtyYEV8iYGTFB7Kc9nvNO4968=">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</latexit><latexit sha1_base64="0jdtyYEV8iYGTFB7Kc9nvNO4968=">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</latexit>

x(1.0s) = 3.0(1) + 0.5(13) = 3.5m<latexit sha1_base64="/QSrqC7/1+j8YAT2lqtY8Ljp3JY=">AAACCnicdVBNS0JBFJ3Xp9nXq5ZtpiRQgseMZuoiENq0NMgPUJN546iD8z6YmReJuG7TX2nTooi2/YJ2/ZvmqUFFHbhwOOde7r3HDQVXGqEPa2FxaXllNbGWXN/Y3Nq2d3ZrKogkZVUaiEA2XKKY4D6raq4Fa4SSEc8VrO4Oz2O/fsOk4oF/pUcha3uk7/Mep0QbqWMf3Kaxg1QGnsGcg9I4c4ycfBpf52ZKHnodO4UchBDGGMYEF06RIaVSMYuLEMeWQQrMUenY761uQCOP+ZoKolQTo1C3x0RqTgWbJFuRYiGhQ9JnTUN94jHVHk9fmcAjo3RhL5CmfA2n6veJMfGUGnmu6fSIHqjfXiz+5TUj3Su2x9wPI818OlvUiwTUAYxzgV0uGdViZAihkptbIR0QSag26SVNCF+fwv9JLetg5ODLk1S5OI8jAfbBIUgDDAqgDC5ABVQBBXfgATyBZ+veerRerNdZ64I1n9kDP2C9fQJSB5Tt</latexit><latexit sha1_base64="/QSrqC7/1+j8YAT2lqtY8Ljp3JY=">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</latexit><latexit sha1_base64="/QSrqC7/1+j8YAT2lqtY8Ljp3JY=">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</latexit><latexit sha1_base64="/QSrqC7/1+j8YAT2lqtY8Ljp3JY=">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</latexit>

x(3.0s) = 3.0(3) + 0.5(33) = 22.5m<latexit sha1_base64="Cm517GcFeQjhGFDGW4/evsbs4kc=">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</latexit><latexit sha1_base64="Cm517GcFeQjhGFDGW4/evsbs4kc=">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</latexit><latexit sha1_base64="Cm517GcFeQjhGFDGW4/evsbs4kc=">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</latexit><latexit sha1_base64="Cm517GcFeQjhGFDGW4/evsbs4kc=">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</latexit>

v =�x

22.5m� 3.5m

3.0s� 1.0s= 9.5m/s

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A. Maller

Rush (2013)

What’s different about the motion in this clip than the first one?

A. Maller

The car goes from 30 to 65 in about 1 second.

Likely it is a European car so we can assume the speedometer is in kilometers per hour.

So thats a change in velocity of 35 km/hr or

35 km/hr × 1000m/km × 1hr/3600s = 9.72m/s

So that’s an acceleration of 9.72m/s / 1s = 9.72 m/s2.

A. Maller

Acceleration

As we’ve seen in many cases the velocity of an object changes, this is called acceleration. The average acceleration over some time period is given by:

and of course the instantaneous acceleration is:

the acceleration can also change with time but we don’t have a new name for that, and constant acceleration is often the case.

a =�v

A. Maller

Example 3.6A particle is in motion and is accelerating. The functional form of the velocity is v(t)=20t−5t2 m/s. a)Find the functional form of the acceleration. b) Find the instantaneous velocity at t = 1, 2, 3, and 5 s. c) Find the instantaneous acceleration at t = 1, 2, 3, and 5 s.

a(t) =dv(t)

dt= 20� 10tm/s2

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b) v(1s)=15m/s, v(2s)=20m/s, v(3s)=15m/s, v(5s)=−25m/s

a) a(1s)=10m/s2, a(2s)=0m/s2, a(3s)=−10m/s2, a(5s)=−30m/s2

A. Maller

Motion at Constant Acceleration

Constant acceleration is very common and so we will show equations of motion in the case of constant acceleration.

Since in this case the acceleration is the average acceleration these can be derived from algebra instead of calculus.

However, you all understand calculus so we will just do it the easy way.

A. Maller

Motion at Constant Acceleration

The equations of motion for constant acceleration are:

v = v0 + at

x = x0 + v0t+1

v2 = v20 + 2a(x� x0)

v =v + v0

We will use these to solve many problems

A. Maller

Deriving the equations of motion

a = constant

Zadt = at+C= v0 + at

Zvdt =

Z(v0 + at)dt = v0t+

2at2 +x0

v = v0 + at => v � v0 = at => t =(v � v0)

x� x0 = v0t+1

2at2 = v0

v � v0a

=> 2a(x� x0) = 2v0(v � v0) + (v � v0)2

= 2v0v � 2v20 + v2 � 2v0v + v20 = v2 � v20

v2 = v20 + 2a(x� x0)

A. Maller

Example 3.6An airplane lands with an initial velocity of 70.0 m/s and then decelerates at 1.50 m/s2 for 40.0 s. What is its final velocity?

known wantx0 = 0v0 = 70.0m/sa = -1.50 m/s2

t = 40.0s

physics?

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equation

plug in

v = v0 + at = 70.0m/s� (1.50m/s2)(40.0s) = 10.0m/s<latexit sha1_base64="nTm8R4TDz/p2Gs2iJUeN2UjfUMw=">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</latexit><latexit sha1_base64="nTm8R4TDz/p2Gs2iJUeN2UjfUMw=">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</latexit><latexit sha1_base64="nTm8R4TDz/p2Gs2iJUeN2UjfUMw=">AAACIXicdZDLSkMxEIZzvFtvVZduBotQEWtS1B4XguDGpYK1hVpLTprWYM6FJKdQiq/ixldx40IRd+LLOMdWUNGBwMf/zzCZP0i0so7SN29sfGJyanpmNjc3v7C4lF9eubBxaoSsiljHph5wK7WKZNUpp2U9MZKHgZa14OY482s9aayKo3PXT2Qz5N1IdZTgDqVW3u/BIfRaFLaAg0Ou0BKFcMfCNhRZaY8iXpU3i7so20302chv5QtIlDLGIANW2acIBwd+mfnAMgurQEZ12sq/XrZjkYYyckJzaxuMJq454MYpoeVt7jK1MuHihndlAzHiobTNweeFt7CBShs6scEXOfhUv08MeGhtPwywM+Tu2v72MvEvr5G6jt8cqChJnYzEcFEn1eBiyOKCtjJSON1H4MIo/CuIa264cBhqDkP4uhT+h4tyCVNjZ7uFI38UxwxZI+ukSBipkCNyQk5JlQhyRx7IE3n27r1H78V7HbaOeaOZVfKjvPcP0iib4g==</latexit><latexit sha1_base64="nTm8R4TDz/p2Gs2iJUeN2UjfUMw=">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</latexit>

A. Maller

Solving Problems1. Draw a Picture

2. List what you are given.

3. Determine what you are supposed to find.

4. Think about the physics that describe the situation

5. Find any relevant equations.

6. Solve for your unknown.

7. Plug in the numbers.

8. Estimate the answer and check the units.

A. Maller

Free Fall

One of the most common examples of constant acceleration is free fall.

Galileo discovered this with his famous experiment dropping two different mass objects from a height.

He observed they reached the ground at the same time, thus they must be accelerating at the same rate.

On the surface of the Earth this constant is given the symbol g and is equal to 9.81 m/s2.

A. Maller

Example 3.14Vertical Motion of a Baseball : A batter hits a baseball straight upward at home plate and the ball is caught 5.0 s after it is struck (a) What is the initial velocity of the ball? (b) What is the maximum height the ball reaches? (c) How long does it take to reach the maximum height? (d) What is the acceleration at the top of its path? (e) What is the velocity of the ball when it is caught? Assume the ball is hit and caught at the same location.

knowny0 = 0

t = 5.0sa = -g = -9.81 m/s2

v0 = ?

negative because opposite direction as v

want physics

ymax = ?tmax = ?amax = ?vf = ?

free fall - constant acceleration

what happens after 5s? y = 0

=> v0 =1

2gt = 0.5(9.8m/s2)(5.0s) = 24.5m/s

<latexit sha1_base64="j0GexITpSbCKl+ec9m84al+UWO8=">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</latexit><latexit sha1_base64="j0GexITpSbCKl+ec9m84al+UWO8=">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</latexit><latexit sha1_base64="j0GexITpSbCKl+ec9m84al+UWO8=">AAACIXicbZDLSgMxFIYz9VbrbdSlm2AR2s2YGVqsi0rBjcsK9gJtLZk004ZmLiSZQhn6Km58FTcuFOlOfBnTC6KtPwR+vnMOJ+d3I86kQujTSG1sbm3vpHcze/sHh0fm8UldhrEgtEZCHoqmiyXlLKA1xRSnzUhQ7LucNtzh7azeGFEhWRg8qHFEOz7uB8xjBCuNumapfANHXQTLsO0JTBJ7kjiTPlQaIKuYu7ZK/qV8dPK5ooVkXlOnYBWhZl0ziyw0F/wx9qrJgqWqXXPa7oUk9mmgCMdStmwUqU6ChWKE00mmHUsaYTLEfdrSNsA+lZ1kfuEEXmjSg14o9AsUnNPfEwn2pRz7ru70sRrI1doM/ldrxcordRIWRLGiAVks8mIOVQhnccEeE5QoPtYGE8H0XyEZYJ2U0qFmdAhrJ6+bumPZyLLvC9lKaRlHGpyBc5ADNrgCFXAHqqAGCHgCL+ANvBvPxqvxYUwXrSljOXMK/sj4+gbFZJ2g</latexit><latexit sha1_base64="j0GexITpSbCKl+ec9m84al+UWO8=">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</latexit>

=> v0t =1

<latexit sha1_base64="D+SjWi0hU3jEKYcJBb7Wh3pLDBA=">AAACBHicbVDLSsNAFJ34rPUVddnNYBFclaQIdlMpuHFZwT6grWEynbRDJ5Mwc1MoIQs3/oobF4q49SPc+TdOH4i2HrhwOOde7r3HjwXX4Dhf1tr6xubWdm4nv7u3f3BoHx03dZQoyho0EpFq+0QzwSVrAAfB2rFiJPQFa/mj66nfGjOleSTvYBKzXkgGkgecEjCSZxeqV3jsOYCruBsoQlM3S8sZHmC4L3t20Sk5M+Af4i6TIlqg7tmf3X5Ek5BJoIJo3XGdGHopUcCpYFm+m2gWEzoiA9YxVJKQ6V46eyLDZ0bp4yBSpiTgmfp7IiWh1pPQN50hgaFe9qbif14ngaDSS7mME2CSzhcFicAQ4WkiuM8VoyAmhhCquLkV0yExWYDJLW9CWHl5lTTLJdcpubcXxVplEUcOFdApOkcuukQ1dIPqqIEoekBP6AW9Wo/Ws/Vmvc9b16zFzAn6A+vjG8Iuljc=</latexit><latexit sha1_base64="D+SjWi0hU3jEKYcJBb7Wh3pLDBA=">AAACBHicbVDLSsNAFJ34rPUVddnNYBFclaQIdlMpuHFZwT6grWEynbRDJ5Mwc1MoIQs3/oobF4q49SPc+TdOH4i2HrhwOOde7r3HjwXX4Dhf1tr6xubWdm4nv7u3f3BoHx03dZQoyho0EpFq+0QzwSVrAAfB2rFiJPQFa/mj66nfGjOleSTvYBKzXkgGkgecEjCSZxeqV3jsOYCruBsoQlM3S8sZHmC4L3t20Sk5M+Af4i6TIlqg7tmf3X5Ek5BJoIJo3XGdGHopUcCpYFm+m2gWEzoiA9YxVJKQ6V46eyLDZ0bp4yBSpiTgmfp7IiWh1pPQN50hgaFe9qbif14ngaDSS7mME2CSzhcFicAQ4WkiuM8VoyAmhhCquLkV0yExWYDJLW9CWHl5lTTLJdcpubcXxVplEUcOFdApOkcuukQ1dIPqqIEoekBP6AW9Wo/Ws/Vmvc9b16zFzAn6A+vjG8Iuljc=</latexit><latexit sha1_base64="D+SjWi0hU3jEKYcJBb7Wh3pLDBA=">AAACBHicbVDLSsNAFJ34rPUVddnNYBFclaQIdlMpuHFZwT6grWEynbRDJ5Mwc1MoIQs3/oobF4q49SPc+TdOH4i2HrhwOOde7r3HjwXX4Dhf1tr6xubWdm4nv7u3f3BoHx03dZQoyho0EpFq+0QzwSVrAAfB2rFiJPQFa/mj66nfGjOleSTvYBKzXkgGkgecEjCSZxeqV3jsOYCruBsoQlM3S8sZHmC4L3t20Sk5M+Af4i6TIlqg7tmf3X5Ek5BJoIJo3XGdGHopUcCpYFm+m2gWEzoiA9YxVJKQ6V46eyLDZ0bp4yBSpiTgmfp7IiWh1pPQN50hgaFe9qbif14ngaDSS7mME2CSzhcFicAQ4WkiuM8VoyAmhhCquLkV0yExWYDJLW9CWHl5lTTLJdcpubcXxVplEUcOFdApOkcuukQ1dIPqqIEoekBP6AW9Wo/Ws/Vmvc9b16zFzAn6A+vjG8Iuljc=</latexit><latexit sha1_base64="D+SjWi0hU3jEKYcJBb7Wh3pLDBA=">AAACBHicbVDLSsNAFJ34rPUVddnNYBFclaQIdlMpuHFZwT6grWEynbRDJ5Mwc1MoIQs3/oobF4q49SPc+TdOH4i2HrhwOOde7r3HjwXX4Dhf1tr6xubWdm4nv7u3f3BoHx03dZQoyho0EpFq+0QzwSVrAAfB2rFiJPQFa/mj66nfGjOleSTvYBKzXkgGkgecEjCSZxeqV3jsOYCruBsoQlM3S8sZHmC4L3t20Sk5M+Af4i6TIlqg7tmf3X5Ek5BJoIJo3XGdGHopUcCpYFm+m2gWEzoiA9YxVJKQ6V46eyLDZ0bp4yBSpiTgmfp7IiWh1pPQN50hgaFe9qbif14ngaDSS7mME2CSzhcFicAQ4WkiuM8VoyAmhhCquLkV0yExWYDJLW9CWHl5lTTLJdcpubcXxVplEUcOFdApOkcuukQ1dIPqqIEoekBP6AW9Wo/Ws/Vmvc9b16zFzAn6A+vjG8Iuljc=</latexit>

=> 0 = 0 + v0t�1

<latexit sha1_base64="W7CypaljTSTi9gJanGVidJ7TN+w=">AAACC3icbVDJSgNBEO1xjXEb9eilSRAEMcwEwVwiAS8eI5gFknHo6fQkTXoWumsCYZi7F3/FiwdFvPoD3vwbOwuiiQ8KHu9VUVXPiwVXYFlfxsrq2vrGZm4rv72zu7dvHhw2VZRIyho0EpFse0QxwUPWAA6CtWPJSOAJ1vKG1xO/NWJS8Si8g3HMnID0Q+5zSkBLrlmoXmELV3Wd4ZFrAT7HXV8SmtpZWs76GO7Lrlm0StYU+IfYi6SI5qi75me3F9EkYCFQQZTq2FYMTkokcCpYlu8misWEDkmfdTQNScCUk05/yfCJVnrYj6SuEPBU/T2RkkCpceDpzoDAQC16E/E/r5OAX3FSHsYJsJDOFvmJwBDhSTC4xyWjIMaaECq5vhXTAdFRgI4vr0NYenmZNMsl2yrZtxfFWmUeRw4dowI6RTa6RDV0g+qogSh6QE/oBb0aj8az8Wa8z1pXjPnMEfoD4+MbjeaXlQ==</latexit><latexit sha1_base64="W7CypaljTSTi9gJanGVidJ7TN+w=">AAACC3icbVDJSgNBEO1xjXEb9eilSRAEMcwEwVwiAS8eI5gFknHo6fQkTXoWumsCYZi7F3/FiwdFvPoD3vwbOwuiiQ8KHu9VUVXPiwVXYFlfxsrq2vrGZm4rv72zu7dvHhw2VZRIyho0EpFse0QxwUPWAA6CtWPJSOAJ1vKG1xO/NWJS8Si8g3HMnID0Q+5zSkBLrlmoXmELV3Wd4ZFrAT7HXV8SmtpZWs76GO7Lrlm0StYU+IfYi6SI5qi75me3F9EkYCFQQZTq2FYMTkokcCpYlu8misWEDkmfdTQNScCUk05/yfCJVnrYj6SuEPBU/T2RkkCpceDpzoDAQC16E/E/r5OAX3FSHsYJsJDOFvmJwBDhSTC4xyWjIMaaECq5vhXTAdFRgI4vr0NYenmZNMsl2yrZtxfFWmUeRw4dowI6RTa6RDV0g+qogSh6QE/oBb0aj8az8Wa8z1pXjPnMEfoD4+MbjeaXlQ==</latexit><latexit sha1_base64="W7CypaljTSTi9gJanGVidJ7TN+w=">AAACC3icbVDJSgNBEO1xjXEb9eilSRAEMcwEwVwiAS8eI5gFknHo6fQkTXoWumsCYZi7F3/FiwdFvPoD3vwbOwuiiQ8KHu9VUVXPiwVXYFlfxsrq2vrGZm4rv72zu7dvHhw2VZRIyho0EpFse0QxwUPWAA6CtWPJSOAJ1vKG1xO/NWJS8Si8g3HMnID0Q+5zSkBLrlmoXmELV3Wd4ZFrAT7HXV8SmtpZWs76GO7Lrlm0StYU+IfYi6SI5qi75me3F9EkYCFQQZTq2FYMTkokcCpYlu8misWEDkmfdTQNScCUk05/yfCJVnrYj6SuEPBU/T2RkkCpceDpzoDAQC16E/E/r5OAX3FSHsYJsJDOFvmJwBDhSTC4xyWjIMaaECq5vhXTAdFRgI4vr0NYenmZNMsl2yrZtxfFWmUeRw4dowI6RTa6RDV0g+qogSh6QE/oBb0aj8az8Wa8z1pXjPnMEfoD4+MbjeaXlQ==</latexit><latexit sha1_base64="W7CypaljTSTi9gJanGVidJ7TN+w=">AAACC3icbVDJSgNBEO1xjXEb9eilSRAEMcwEwVwiAS8eI5gFknHo6fQkTXoWumsCYZi7F3/FiwdFvPoD3vwbOwuiiQ8KHu9VUVXPiwVXYFlfxsrq2vrGZm4rv72zu7dvHhw2VZRIyho0EpFse0QxwUPWAA6CtWPJSOAJ1vKG1xO/NWJS8Si8g3HMnID0Q+5zSkBLrlmoXmELV3Wd4ZFrAT7HXV8SmtpZWs76GO7Lrlm0StYU+IfYi6SI5qi75me3F9EkYCFQQZTq2FYMTkokcCpYlu8misWEDkmfdTQNScCUk05/yfCJVnrYj6SuEPBU/T2RkkCpceDpzoDAQC16E/E/r5OAX3FSHsYJsJDOFvmJwBDhSTC4xyWjIMaaECq5vhXTAdFRgI4vr0NYenmZNMsl2yrZtxfFWmUeRw4dowI6RTa6RDV0g+qogSh6QE/oBb0aj8az8Wa8z1pXjPnMEfoD4+MbjeaXlQ==</latexit>

y = y0 + v0t�1

<latexit sha1_base64="7ZyYcrDpS21KPmQDv7fltIOXBH8=">AAACDHicbVDLSgMxFM3UV62vqks3wSIIYpkpgt0IBTcuK9gHtGPJpJk2NJMZkjuFYZgPcOOvuHGhiFs/wJ1/Y/pAtPVAwuGcc0nu8SLBNdj2l5VbWV1b38hvFra2d3b3ivsHTR3GirIGDUWo2h7RTHDJGsBBsHakGAk8wVre6Hrit8ZMaR7KO0gi5gZkILnPKQEj9YqlBF/hpGfjMzw2N+Bz3PUVoamTpZUMDzDcV0zKLttT4B/iLJISmqPeK352+yGNAyaBCqJ1x7EjcFOigFPBskI31iwidEQGrGOoJAHTbjpdJsMnRuljP1TmSMBT9fdESgKtk8AzyYDAUC96E/E/rxODX3VTLqMYmKSzh/xYYAjxpBnc54pREIkhhCpu/orpkJguwPRXMCUsrbxMmpWyY5ed24tSrTqvI4+O0DE6RQ66RDV0g+qogSh6QE/oBb1aj9az9Wa9z6I5az5ziP7A+vgG/NiYZQ==</latexit><latexit sha1_base64="7ZyYcrDpS21KPmQDv7fltIOXBH8=">AAACDHicbVDLSgMxFM3UV62vqks3wSIIYpkpgt0IBTcuK9gHtGPJpJk2NJMZkjuFYZgPcOOvuHGhiFs/wJ1/Y/pAtPVAwuGcc0nu8SLBNdj2l5VbWV1b38hvFra2d3b3ivsHTR3GirIGDUWo2h7RTHDJGsBBsHakGAk8wVre6Hrit8ZMaR7KO0gi5gZkILnPKQEj9YqlBF/hpGfjMzw2N+Bz3PUVoamTpZUMDzDcV0zKLttT4B/iLJISmqPeK352+yGNAyaBCqJ1x7EjcFOigFPBskI31iwidEQGrGOoJAHTbjpdJsMnRuljP1TmSMBT9fdESgKtk8AzyYDAUC96E/E/rxODX3VTLqMYmKSzh/xYYAjxpBnc54pREIkhhCpu/orpkJguwPRXMCUsrbxMmpWyY5ed24tSrTqvI4+O0DE6RQ66RDV0g+qogSh6QE/oBb1aj9az9Wa9z6I5az5ziP7A+vgG/NiYZQ==</latexit><latexit sha1_base64="7ZyYcrDpS21KPmQDv7fltIOXBH8=">AAACDHicbVDLSgMxFM3UV62vqks3wSIIYpkpgt0IBTcuK9gHtGPJpJk2NJMZkjuFYZgPcOOvuHGhiFs/wJ1/Y/pAtPVAwuGcc0nu8SLBNdj2l5VbWV1b38hvFra2d3b3ivsHTR3GirIGDUWo2h7RTHDJGsBBsHakGAk8wVre6Hrit8ZMaR7KO0gi5gZkILnPKQEj9YqlBF/hpGfjMzw2N+Bz3PUVoamTpZUMDzDcV0zKLttT4B/iLJISmqPeK352+yGNAyaBCqJ1x7EjcFOigFPBskI31iwidEQGrGOoJAHTbjpdJsMnRuljP1TmSMBT9fdESgKtk8AzyYDAUC96E/E/rxODX3VTLqMYmKSzh/xYYAjxpBnc54pREIkhhCpu/orpkJguwPRXMCUsrbxMmpWyY5ed24tSrTqvI4+O0DE6RQ66RDV0g+qogSh6QE/oBb1aj9az9Wa9z6I5az5ziP7A+vgG/NiYZQ==</latexit><latexit sha1_base64="7ZyYcrDpS21KPmQDv7fltIOXBH8=">AAACDHicbVDLSgMxFM3UV62vqks3wSIIYpkpgt0IBTcuK9gHtGPJpJk2NJMZkjuFYZgPcOOvuHGhiFs/wJ1/Y/pAtPVAwuGcc0nu8SLBNdj2l5VbWV1b38hvFra2d3b3ivsHTR3GirIGDUWo2h7RTHDJGsBBsHakGAk8wVre6Hrit8ZMaR7KO0gi5gZkILnPKQEj9YqlBF/hpGfjMzw2N+Bz3PUVoamTpZUMDzDcV0zKLttT4B/iLJISmqPeK352+yGNAyaBCqJ1x7EjcFOigFPBskI31iwidEQGrGOoJAHTbjpdJsMnRuljP1TmSMBT9fdESgKtk8AzyYDAUC96E/E/rxODX3VTLqMYmKSzh/xYYAjxpBnc54pREIkhhCpu/orpkJguwPRXMCUsrbxMmpWyY5ed24tSrTqvI4+O0DE6RQ66RDV0g+qogSh6QE/oBb1aj9az9Wa9z6I5az5ziP7A+vgG/NiYZQ==</latexit>

what happens at ymax? v = 0 v2 = v20 � 2g(ymax � y0)<latexit sha1_base64="BJ+Vcw+vhlt/mhffApLWVkaFOPU=">AAACBnicbVDLSgMxFM3UV62vUZciBItQFy2ZItiNUHDjsoJ9QDsOmTRtQ5OZIckUh6ErN/6KGxeKuPUb3Pk3pg9EWw9c7uGce0nu8SPOlEboy8qsrK6tb2Q3c1vbO7t79v5BQ4WxJLROQh7Klo8V5Sygdc00p61IUix8Tpv+8GriN0dUKhYGtzqJqCtwP2A9RrA2kmcfj+7K8BKOPGR6EZb7hcRLBb4fFxMPnXl2HpXQFPCHOIskD+aoefZnpxuSWNBAE46Vajso0m6KpWaE03GuEysaYTLEfdo2NMCCKjednjGGp0bpwl4oTQUaTtXfGykWSiXCN5MC64Fa9Cbif1471r2Km7IgijUNyOyhXsyhDuEkE9hlkhLNE0Mwkcz8FZIBlphok1zOhLB08jJplEsOKjk35/lqZR5HFhyBE1AADrgAVXANaqAOCHgAT+AFvFqP1rP1Zr3PRjPWfOcQ/IH18Q1jEZZ+</latexit><latexit sha1_base64="BJ+Vcw+vhlt/mhffApLWVkaFOPU=">AAACBnicbVDLSgMxFM3UV62vUZciBItQFy2ZItiNUHDjsoJ9QDsOmTRtQ5OZIckUh6ErN/6KGxeKuPUb3Pk3pg9EWw9c7uGce0nu8SPOlEboy8qsrK6tb2Q3c1vbO7t79v5BQ4WxJLROQh7Klo8V5Sygdc00p61IUix8Tpv+8GriN0dUKhYGtzqJqCtwP2A9RrA2kmcfj+7K8BKOPGR6EZb7hcRLBb4fFxMPnXl2HpXQFPCHOIskD+aoefZnpxuSWNBAE46Vajso0m6KpWaE03GuEysaYTLEfdo2NMCCKjednjGGp0bpwl4oTQUaTtXfGykWSiXCN5MC64Fa9Cbif1471r2Km7IgijUNyOyhXsyhDuEkE9hlkhLNE0Mwkcz8FZIBlphok1zOhLB08jJplEsOKjk35/lqZR5HFhyBE1AADrgAVXANaqAOCHgAT+AFvFqP1rP1Zr3PRjPWfOcQ/IH18Q1jEZZ+</latexit><latexit sha1_base64="BJ+Vcw+vhlt/mhffApLWVkaFOPU=">AAACBnicbVDLSgMxFM3UV62vUZciBItQFy2ZItiNUHDjsoJ9QDsOmTRtQ5OZIckUh6ErN/6KGxeKuPUb3Pk3pg9EWw9c7uGce0nu8SPOlEboy8qsrK6tb2Q3c1vbO7t79v5BQ4WxJLROQh7Klo8V5Sygdc00p61IUix8Tpv+8GriN0dUKhYGtzqJqCtwP2A9RrA2kmcfj+7K8BKOPGR6EZb7hcRLBb4fFxMPnXl2HpXQFPCHOIskD+aoefZnpxuSWNBAE46Vajso0m6KpWaE03GuEysaYTLEfdo2NMCCKjednjGGp0bpwl4oTQUaTtXfGykWSiXCN5MC64Fa9Cbif1471r2Km7IgijUNyOyhXsyhDuEkE9hlkhLNE0Mwkcz8FZIBlphok1zOhLB08jJplEsOKjk35/lqZR5HFhyBE1AADrgAVXANaqAOCHgAT+AFvFqP1rP1Zr3PRjPWfOcQ/IH18Q1jEZZ+</latexit><latexit sha1_base64="BJ+Vcw+vhlt/mhffApLWVkaFOPU=">AAACBnicbVDLSgMxFM3UV62vUZciBItQFy2ZItiNUHDjsoJ9QDsOmTRtQ5OZIckUh6ErN/6KGxeKuPUb3Pk3pg9EWw9c7uGce0nu8SPOlEboy8qsrK6tb2Q3c1vbO7t79v5BQ4WxJLROQh7Klo8V5Sygdc00p61IUix8Tpv+8GriN0dUKhYGtzqJqCtwP2A9RrA2kmcfj+7K8BKOPGR6EZb7hcRLBb4fFxMPnXl2HpXQFPCHOIskD+aoefZnpxuSWNBAE46Vajso0m6KpWaE03GuEysaYTLEfdo2NMCCKjednjGGp0bpwl4oTQUaTtXfGykWSiXCN5MC64Fa9Cbif1471r2Km7IgijUNyOyhXsyhDuEkE9hlkhLNE0Mwkcz8FZIBlphok1zOhLB08jJplEsOKjk35/lqZR5HFhyBE1AADrgAVXANaqAOCHgAT+AFvFqP1rP1Zr3PRjPWfOcQ/IH18Q1jEZZ+</latexit>

=> v20 = 2gymax<latexit sha1_base64="rpWhoIJSldcYiijOP9kFCr4O8Qg=">AAAB/nicbVDLSsNAFL2pr1pfUXHlZrAIrkpSBLupFNy4rGAfUGOYTKft0MkkzEyKIRT8FTcuFHHrd7jzb5w+EG09cOFwzr3ce08Qc6a043xZuZXVtfWN/GZha3tnd8/eP2iqKJGENkjEI9kOsKKcCdrQTHPajiXFYcBpKxheTfzWiErFInGr05h6Ie4L1mMEayP59lH1Eo18576MqqjcR6mfhfhh7NtFp+RMgX6Iu0iKMEfdtz/vuhFJQio04VipjuvE2suw1IxwOi7cJYrGmAxxn3YMFTikysum54/RqVG6qBdJU0Kjqfp7IsOhUmkYmM4Q64Fa9Cbif14n0b2KlzERJ5oKMlvUSzjSEZpkgbpMUqJ5aggmkplbERlgiYk2iRVMCEsvL5NmueQ6JffmvFirzOPIwzGcwBm4cAE1uIY6NIBABk/wAq/Wo/VsvVnvs9acNZ85hD+wPr4Bl2yT8A==</latexit><latexit sha1_base64="rpWhoIJSldcYiijOP9kFCr4O8Qg=">AAAB/nicbVDLSsNAFL2pr1pfUXHlZrAIrkpSBLupFNy4rGAfUGOYTKft0MkkzEyKIRT8FTcuFHHrd7jzb5w+EG09cOFwzr3ce08Qc6a043xZuZXVtfWN/GZha3tnd8/eP2iqKJGENkjEI9kOsKKcCdrQTHPajiXFYcBpKxheTfzWiErFInGr05h6Ie4L1mMEayP59lH1Eo18576MqqjcR6mfhfhh7NtFp+RMgX6Iu0iKMEfdtz/vuhFJQio04VipjuvE2suw1IxwOi7cJYrGmAxxn3YMFTikysum54/RqVG6qBdJU0Kjqfp7IsOhUmkYmM4Q64Fa9Cbif14n0b2KlzERJ5oKMlvUSzjSEZpkgbpMUqJ5aggmkplbERlgiYk2iRVMCEsvL5NmueQ6JffmvFirzOPIwzGcwBm4cAE1uIY6NIBABk/wAq/Wo/VsvVnvs9acNZ85hD+wPr4Bl2yT8A==</latexit><latexit sha1_base64="rpWhoIJSldcYiijOP9kFCr4O8Qg=">AAAB/nicbVDLSsNAFL2pr1pfUXHlZrAIrkpSBLupFNy4rGAfUGOYTKft0MkkzEyKIRT8FTcuFHHrd7jzb5w+EG09cOFwzr3ce08Qc6a043xZuZXVtfWN/GZha3tnd8/eP2iqKJGENkjEI9kOsKKcCdrQTHPajiXFYcBpKxheTfzWiErFInGr05h6Ie4L1mMEayP59lH1Eo18576MqqjcR6mfhfhh7NtFp+RMgX6Iu0iKMEfdtz/vuhFJQio04VipjuvE2suw1IxwOi7cJYrGmAxxn3YMFTikysum54/RqVG6qBdJU0Kjqfp7IsOhUmkYmM4Q64Fa9Cbif14n0b2KlzERJ5oKMlvUSzjSEZpkgbpMUqJ5aggmkplbERlgiYk2iRVMCEsvL5NmueQ6JffmvFirzOPIwzGcwBm4cAE1uIY6NIBABk/wAq/Wo/VsvVnvs9acNZ85hD+wPr4Bl2yT8A==</latexit><latexit sha1_base64="rpWhoIJSldcYiijOP9kFCr4O8Qg=">AAAB/nicbVDLSsNAFL2pr1pfUXHlZrAIrkpSBLupFNy4rGAfUGOYTKft0MkkzEyKIRT8FTcuFHHrd7jzb5w+EG09cOFwzr3ce08Qc6a043xZuZXVtfWN/GZha3tnd8/eP2iqKJGENkjEI9kOsKKcCdrQTHPajiXFYcBpKxheTfzWiErFInGr05h6Ie4L1mMEayP59lH1Eo18576MqqjcR6mfhfhh7NtFp+RMgX6Iu0iKMEfdtz/vuhFJQio04VipjuvE2suw1IxwOi7cJYrGmAxxn3YMFTikysum54/RqVG6qBdJU0Kjqfp7IsOhUmkYmM4Q64Fa9Cbif14n0b2KlzERJ5oKMlvUSzjSEZpkgbpMUqJ5aggmkplbERlgiYk2iRVMCEsvL5NmueQ6JffmvFirzOPIwzGcwBm4cAE1uIY6NIBABk/wAq/Wo/VsvVnvs9acNZ85hD+wPr4Bl2yT8A==</latexit>

=> 2gymax = v20/2g =(24.5m/s)2

2 ⇤ 9.8m/s2= 30.6m

<latexit sha1_base64="eQ6HM6nVzDT5ghs0X8iIABMJx0U=">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</latexit><latexit sha1_base64="eQ6HM6nVzDT5ghs0X8iIABMJx0U=">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</latexit><latexit sha1_base64="eQ6HM6nVzDT5ghs0X8iIABMJx0U=">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</latexit><latexit sha1_base64="eQ6HM6nVzDT5ghs0X8iIABMJx0U=">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</latexit>

A. Maller

Example 3.14Vertical Motion of a Baseball : A batter hits a baseball straight upward at home plate and the ball is caught 5.0 s after it is struck (a) What is the initial velocity of the ball? (b) What is the maximum height the ball reaches? (c) How long does it take to reach the maximum height? (d) What is the acceleration at the top of its path? (e) What is the velocity of the ball when it is caught? Assume the ball is hit and caught at the same location.

knowny0 = 0

t = 5.0sa = -g = -9.81 m/s2

v0 = ?

negative because opposite direction as v

want physics

ymax = ?tmax = ?amax = ?vf = ?

tmax y is ymax

what happens at ymax? v = 0

v = v0 � gt<latexit sha1_base64="lXqeYsZh0Rhb9H55GZiwe3AYyK4=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgxbArgrkIAS8eI5gHJEuYnUySIbOz60zvQljyHV48KOLVj/Hm3zh5IJpY0FBUddPdFcRSGHTdLye3tr6xuZXfLuzs7u0fFA+PGiZKNON1FslItwJquBSK11Gg5K1YcxoGkjeD0e3Ub6ZcGxGpBxzH3A/pQIm+YBSt5KfkhqRdl1yQAcFuseSW3RnID/GWSQkWqHWLn51exJKQK2SSGtP23Bj9jGoUTPJJoZMYHlM2ogPetlTRkBs/mx09IWdW6ZF+pG0pJDP190RGQ2PGYWA7Q4pDs+xNxf+8doL9ip8JFSfIFZsv6ieSYESmCZCe0JyhHFtCmRb2VsKGVFOGNqeCDWHl5VXSuCx7btm7vypVK4s48nACp3AOHlxDFe6gBnVg8AhP8AKvTuo8O2/O+7w15yxmjuEPnI9vHFeQUw==</latexit><latexit sha1_base64="lXqeYsZh0Rhb9H55GZiwe3AYyK4=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgxbArgrkIAS8eI5gHJEuYnUySIbOz60zvQljyHV48KOLVj/Hm3zh5IJpY0FBUddPdFcRSGHTdLye3tr6xuZXfLuzs7u0fFA+PGiZKNON1FslItwJquBSK11Gg5K1YcxoGkjeD0e3Ub6ZcGxGpBxzH3A/pQIm+YBSt5KfkhqRdl1yQAcFuseSW3RnID/GWSQkWqHWLn51exJKQK2SSGtP23Bj9jGoUTPJJoZMYHlM2ogPetlTRkBs/mx09IWdW6ZF+pG0pJDP190RGQ2PGYWA7Q4pDs+xNxf+8doL9ip8JFSfIFZsv6ieSYESmCZCe0JyhHFtCmRb2VsKGVFOGNqeCDWHl5VXSuCx7btm7vypVK4s48nACp3AOHlxDFe6gBnVg8AhP8AKvTuo8O2/O+7w15yxmjuEPnI9vHFeQUw==</latexit><latexit sha1_base64="lXqeYsZh0Rhb9H55GZiwe3AYyK4=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgxbArgrkIAS8eI5gHJEuYnUySIbOz60zvQljyHV48KOLVj/Hm3zh5IJpY0FBUddPdFcRSGHTdLye3tr6xuZXfLuzs7u0fFA+PGiZKNON1FslItwJquBSK11Gg5K1YcxoGkjeD0e3Ub6ZcGxGpBxzH3A/pQIm+YBSt5KfkhqRdl1yQAcFuseSW3RnID/GWSQkWqHWLn51exJKQK2SSGtP23Bj9jGoUTPJJoZMYHlM2ogPetlTRkBs/mx09IWdW6ZF+pG0pJDP190RGQ2PGYWA7Q4pDs+xNxf+8doL9ip8JFSfIFZsv6ieSYESmCZCe0JyhHFtCmRb2VsKGVFOGNqeCDWHl5VXSuCx7btm7vypVK4s48nACp3AOHlxDFe6gBnVg8AhP8AKvTuo8O2/O+7w15yxmjuEPnI9vHFeQUw==</latexit><latexit sha1_base64="lXqeYsZh0Rhb9H55GZiwe3AYyK4=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgxbArgrkIAS8eI5gHJEuYnUySIbOz60zvQljyHV48KOLVj/Hm3zh5IJpY0FBUddPdFcRSGHTdLye3tr6xuZXfLuzs7u0fFA+PGiZKNON1FslItwJquBSK11Gg5K1YcxoGkjeD0e3Ub6ZcGxGpBxzH3A/pQIm+YBSt5KfkhqRdl1yQAcFuseSW3RnID/GWSQkWqHWLn51exJKQK2SSGtP23Bj9jGoUTPJJoZMYHlM2ogPetlTRkBs/mx09IWdW6ZF+pG0pJDP190RGQ2PGYWA7Q4pDs+xNxf+8doL9ip8JFSfIFZsv6ieSYESmCZCe0JyhHFtCmRb2VsKGVFOGNqeCDWHl5VXSuCx7btm7vypVK4s48nACp3AOHlxDFe6gBnVg8AhP8AKvTuo8O2/O+7w15yxmjuEPnI9vHFeQUw==</latexit>

=> v0 = gt<latexit sha1_base64="BbHTCNnAMVsxBif8KgfTy7uEzxM=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoPgKeyKYC6RgBePEcwDkiXMTmaTIbOzy0xvICz5DS8eFPHqz3jzb5w8EE0saCiquunuChIpDLrul5Pb2Nza3snvFvb2Dw6PiscnTROnmvEGi2Ws2wE1XArFGyhQ8naiOY0CyVvB6G7mt8ZcGxGrR5wk3I/oQIlQMIpW6lZvybjnkioZEOwVS27ZnYP8EG+VlGCJeq/42e3HLI24QiapMR3PTdDPqEbBJJ8WuqnhCWUjOuAdSxWNuPGz+c1TcmGVPgljbUshmau/JzIaGTOJAtsZURyaVW8m/ud1UgwrfiZUkiJXbLEoTCXBmMwCIH2hOUM5sYQyLeythA2ppgxtTAUbwtrL66R5VfbcsvdwXapVlnHk4QzO4RI8uIEa3EMdGsAggSd4gVcndZ6dN+d90ZpzljOn8AfOxzeGL5AB</latexit><latexit sha1_base64="BbHTCNnAMVsxBif8KgfTy7uEzxM=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoPgKeyKYC6RgBePEcwDkiXMTmaTIbOzy0xvICz5DS8eFPHqz3jzb5w8EE0saCiquunuChIpDLrul5Pb2Nza3snvFvb2Dw6PiscnTROnmvEGi2Ws2wE1XArFGyhQ8naiOY0CyVvB6G7mt8ZcGxGrR5wk3I/oQIlQMIpW6lZvybjnkioZEOwVS27ZnYP8EG+VlGCJeq/42e3HLI24QiapMR3PTdDPqEbBJJ8WuqnhCWUjOuAdSxWNuPGz+c1TcmGVPgljbUshmau/JzIaGTOJAtsZURyaVW8m/ud1UgwrfiZUkiJXbLEoTCXBmMwCIH2hOUM5sYQyLeythA2ppgxtTAUbwtrL66R5VfbcsvdwXapVlnHk4QzO4RI8uIEa3EMdGsAggSd4gVcndZ6dN+d90ZpzljOn8AfOxzeGL5AB</latexit><latexit sha1_base64="BbHTCNnAMVsxBif8KgfTy7uEzxM=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoPgKeyKYC6RgBePEcwDkiXMTmaTIbOzy0xvICz5DS8eFPHqz3jzb5w8EE0saCiquunuChIpDLrul5Pb2Nza3snvFvb2Dw6PiscnTROnmvEGi2Ws2wE1XArFGyhQ8naiOY0CyVvB6G7mt8ZcGxGrR5wk3I/oQIlQMIpW6lZvybjnkioZEOwVS27ZnYP8EG+VlGCJeq/42e3HLI24QiapMR3PTdDPqEbBJJ8WuqnhCWUjOuAdSxWNuPGz+c1TcmGVPgljbUshmau/JzIaGTOJAtsZURyaVW8m/ud1UgwrfiZUkiJXbLEoTCXBmMwCIH2hOUM5sYQyLeythA2ppgxtTAUbwtrL66R5VfbcsvdwXapVlnHk4QzO4RI8uIEa3EMdGsAggSd4gVcndZ6dN+d90ZpzljOn8AfOxzeGL5AB</latexit><latexit sha1_base64="BbHTCNnAMVsxBif8KgfTy7uEzxM=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoPgKeyKYC6RgBePEcwDkiXMTmaTIbOzy0xvICz5DS8eFPHqz3jzb5w8EE0saCiquunuChIpDLrul5Pb2Nza3snvFvb2Dw6PiscnTROnmvEGi2Ws2wE1XArFGyhQ8naiOY0CyVvB6G7mt8ZcGxGrR5wk3I/oQIlQMIpW6lZvybjnkioZEOwVS27ZnYP8EG+VlGCJeq/42e3HLI24QiapMR3PTdDPqEbBJJ8WuqnhCWUjOuAdSxWNuPGz+c1TcmGVPgljbUshmau/JzIaGTOJAtsZURyaVW8m/ud1UgwrfiZUkiJXbLEoTCXBmMwCIH2hOUM5sYQyLeythA2ppgxtTAUbwtrL66R5VfbcsvdwXapVlnHk4QzO4RI8uIEa3EMdGsAggSd4gVcndZ6dN+d90ZpzljOn8AfOxzeGL5AB</latexit>

=> t = v0/g =24.5m/s

9.8m/s2= 2.5s

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notice this t isn’t the same as before, better to call this tmax and the first tf

what is amax? constant acceleration, a always has the same value amax = a = �9.8m/s2

<latexit sha1_base64="+RCL9ffeL+IzJttIHwP+N04QFko=">AAACAHicbVDLSgNBEJyNrxhfqx48eBkMghfjbhCMByHgxWME84Akht7JbDJkZnaZmRXDkou/4sWDIl79DG/+jZMHotGChqKqm+6uIOZMG8/7dDILi0vLK9nV3Nr6xuaWu71T01GiCK2SiEeqEYCmnElaNcxw2ogVBRFwWg8Gl2O/fkeVZpG8McOYtgX0JAsZAWOljrsHnVTA/QhfYLB1fF4oiRN9W+y4ea/gTYC/iT9P8miGSsf9aHUjkggqDeGgddP3YtNOQRlGOB3lWommMZAB9GjTUgmC6nY6eWCED63SxWGkbEmDJ+rPiRSE1kMR2E4Bpq/nvbH4n9dMTFhqp0zGiaGSTBeFCccmwuM0cJcpSgwfWgJEMXsrJn1QQIzNLGdD+PPyX1IrFnyv4F+f5sulWRxZtI8O0BHy0RkqoytUQVVE0Ag9omf04jw4T86r8zZtzTizmV30C877F2gUlEw=</latexit><latexit sha1_base64="+RCL9ffeL+IzJttIHwP+N04QFko=">AAACAHicbVDLSgNBEJyNrxhfqx48eBkMghfjbhCMByHgxWME84Akht7JbDJkZnaZmRXDkou/4sWDIl79DG/+jZMHotGChqKqm+6uIOZMG8/7dDILi0vLK9nV3Nr6xuaWu71T01GiCK2SiEeqEYCmnElaNcxw2ogVBRFwWg8Gl2O/fkeVZpG8McOYtgX0JAsZAWOljrsHnVTA/QhfYLB1fF4oiRN9W+y4ea/gTYC/iT9P8miGSsf9aHUjkggqDeGgddP3YtNOQRlGOB3lWommMZAB9GjTUgmC6nY6eWCED63SxWGkbEmDJ+rPiRSE1kMR2E4Bpq/nvbH4n9dMTFhqp0zGiaGSTBeFCccmwuM0cJcpSgwfWgJEMXsrJn1QQIzNLGdD+PPyX1IrFnyv4F+f5sulWRxZtI8O0BHy0RkqoytUQVVE0Ag9omf04jw4T86r8zZtzTizmV30C877F2gUlEw=</latexit><latexit sha1_base64="+RCL9ffeL+IzJttIHwP+N04QFko=">AAACAHicbVDLSgNBEJyNrxhfqx48eBkMghfjbhCMByHgxWME84Akht7JbDJkZnaZmRXDkou/4sWDIl79DG/+jZMHotGChqKqm+6uIOZMG8/7dDILi0vLK9nV3Nr6xuaWu71T01GiCK2SiEeqEYCmnElaNcxw2ogVBRFwWg8Gl2O/fkeVZpG8McOYtgX0JAsZAWOljrsHnVTA/QhfYLB1fF4oiRN9W+y4ea/gTYC/iT9P8miGSsf9aHUjkggqDeGgddP3YtNOQRlGOB3lWommMZAB9GjTUgmC6nY6eWCED63SxWGkbEmDJ+rPiRSE1kMR2E4Bpq/nvbH4n9dMTFhqp0zGiaGSTBeFCccmwuM0cJcpSgwfWgJEMXsrJn1QQIzNLGdD+PPyX1IrFnyv4F+f5sulWRxZtI8O0BHy0RkqoytUQVVE0Ag9omf04jw4T86r8zZtzTizmV30C877F2gUlEw=</latexit><latexit sha1_base64="+RCL9ffeL+IzJttIHwP+N04QFko=">AAACAHicbVDLSgNBEJyNrxhfqx48eBkMghfjbhCMByHgxWME84Akht7JbDJkZnaZmRXDkou/4sWDIl79DG/+jZMHotGChqKqm+6uIOZMG8/7dDILi0vLK9nV3Nr6xuaWu71T01GiCK2SiEeqEYCmnElaNcxw2ogVBRFwWg8Gl2O/fkeVZpG8McOYtgX0JAsZAWOljrsHnVTA/QhfYLB1fF4oiRN9W+y4ea/gTYC/iT9P8miGSsf9aHUjkggqDeGgddP3YtNOQRlGOB3lWommMZAB9GjTUgmC6nY6eWCED63SxWGkbEmDJ+rPiRSE1kMR2E4Bpq/nvbH4n9dMTFhqp0zGiaGSTBeFCccmwuM0cJcpSgwfWgJEMXsrJn1QQIzNLGdD+PPyX1IrFnyv4F+f5sulWRxZtI8O0BHy0RkqoytUQVVE0Ag9omf04jw4T86r8zZtzTizmV30C877F2gUlEw=</latexit>v = v0 � gt = 24.5m/s� (9.8m/s2)(5s) = �24.5m/s<latexit sha1_base64="QtBZeE7CmQ//BaKQjs1H3DiQbIA=">AAACG3icbZDLSgMxFIYz9VbrbdSlm2AR2kXHmaHFuhAKblxWsBdoxyGTpm1o5kKSKZSh7+HGV3HjQhFXggvfxrQdRFsPBL78/zkk5/ciRoU0zS8ts7a+sbmV3c7t7O7tH+iHR00RxhyTBg5ZyNseEoTRgDQklYy0I06Q7zHS8kbXM781JlzQMLiTk4g4PhoEtE8xkkpydXsMr+DYNWEJDqRCu2xU/HOhroVLo6ro3i4WKqKorFLquXreNMx5wR+wliEP0qq7+ke3F+LYJ4HEDAnRscxIOgnikmJGprluLEiE8AgNSEdhgHwinGS+2xSeKaUH+yFXJ5Bwrv6eSJAvxMT3VKeP5FAsezPxP68Ty37VSWgQxZIEePFQP2ZQhnAWFOxRTrBkEwUIc6r+CvEQcYSlijOnQlhZeRWatmGZhnVbzteqaRxZcAJOQQFY4ALUwA2ogwbA4AE8gRfwqj1qz9qb9r5ozWjpzDH4U9rnNzCumqI=</latexit><latexit sha1_base64="QtBZeE7CmQ//BaKQjs1H3DiQbIA=">AAACG3icbZDLSgMxFIYz9VbrbdSlm2AR2kXHmaHFuhAKblxWsBdoxyGTpm1o5kKSKZSh7+HGV3HjQhFXggvfxrQdRFsPBL78/zkk5/ciRoU0zS8ts7a+sbmV3c7t7O7tH+iHR00RxhyTBg5ZyNseEoTRgDQklYy0I06Q7zHS8kbXM781JlzQMLiTk4g4PhoEtE8xkkpydXsMr+DYNWEJDqRCu2xU/HOhroVLo6ro3i4WKqKorFLquXreNMx5wR+wliEP0qq7+ke3F+LYJ4HEDAnRscxIOgnikmJGprluLEiE8AgNSEdhgHwinGS+2xSeKaUH+yFXJ5Bwrv6eSJAvxMT3VKeP5FAsezPxP68Ty37VSWgQxZIEePFQP2ZQhnAWFOxRTrBkEwUIc6r+CvEQcYSlijOnQlhZeRWatmGZhnVbzteqaRxZcAJOQQFY4ALUwA2ogwbA4AE8gRfwqj1qz9qb9r5ozWjpzDH4U9rnNzCumqI=</latexit><latexit sha1_base64="QtBZeE7CmQ//BaKQjs1H3DiQbIA=">AAACG3icbZDLSgMxFIYz9VbrbdSlm2AR2kXHmaHFuhAKblxWsBdoxyGTpm1o5kKSKZSh7+HGV3HjQhFXggvfxrQdRFsPBL78/zkk5/ciRoU0zS8ts7a+sbmV3c7t7O7tH+iHR00RxhyTBg5ZyNseEoTRgDQklYy0I06Q7zHS8kbXM781JlzQMLiTk4g4PhoEtE8xkkpydXsMr+DYNWEJDqRCu2xU/HOhroVLo6ro3i4WKqKorFLquXreNMx5wR+wliEP0qq7+ke3F+LYJ4HEDAnRscxIOgnikmJGprluLEiE8AgNSEdhgHwinGS+2xSeKaUH+yFXJ5Bwrv6eSJAvxMT3VKeP5FAsezPxP68Ty37VSWgQxZIEePFQP2ZQhnAWFOxRTrBkEwUIc6r+CvEQcYSlijOnQlhZeRWatmGZhnVbzteqaRxZcAJOQQFY4ALUwA2ogwbA4AE8gRfwqj1qz9qb9r5ozWjpzDH4U9rnNzCumqI=</latexit><latexit sha1_base64="QtBZeE7CmQ//BaKQjs1H3DiQbIA=">AAACG3icbZDLSgMxFIYz9VbrbdSlm2AR2kXHmaHFuhAKblxWsBdoxyGTpm1o5kKSKZSh7+HGV3HjQhFXggvfxrQdRFsPBL78/zkk5/ciRoU0zS8ts7a+sbmV3c7t7O7tH+iHR00RxhyTBg5ZyNseEoTRgDQklYy0I06Q7zHS8kbXM781JlzQMLiTk4g4PhoEtE8xkkpydXsMr+DYNWEJDqRCu2xU/HOhroVLo6ro3i4WKqKorFLquXreNMx5wR+wliEP0qq7+ke3F+LYJ4HEDAnRscxIOgnikmJGprluLEiE8AgNSEdhgHwinGS+2xSeKaUH+yFXJ5Bwrv6eSJAvxMT3VKeP5FAsezPxP68Ty37VSWgQxZIEePFQP2ZQhnAWFOxRTrBkEwUIc6r+CvEQcYSlijOnQlhZeRWatmGZhnVbzteqaRxZcAJOQQFY4ALUwA2ogwbA4AE8gRfwqj1qz9qb9r5ozWjpzDH4U9rnNzCumqI=</latexit>

orv2 = v20 � 2g(y � y0)

<latexit sha1_base64="8yflH6XmxeFIYfP+0DsvkgYfluk=">AAACAHicbVDLSgMxFM3UV62vURcu3ASLUBctmSLYjVBw47KCfUA7Dpk004ZmMkOSKQxDN/6KGxeKuPUz3Pk3pg9EWw9c7uGce0nu8WPOlEboy8qtrW9sbuW3Czu7e/sH9uFRS0WJJLRJIh7Jjo8V5UzQpmaa004sKQ59Ttv+6Gbqt8dUKhaJe53G1A3xQLCAEayN5Nkn44cqvIZjD5lehtVBKS2nHrrw7CKqoBngD3GWSREs0PDsz14/IklIhSYcK9V1UKzdDEvNCKeTQi9RNMZkhAe0a6jAIVVuNjtgAs+N0odBJE0JDWfq740Mh0qloW8mQ6yHatmbiv953UQHNTdjIk40FWT+UJBwqCM4TQP2maRE89QQTCQzf4VkiCUm2mRWMCGsnLxKWtWKgyrO3WWxXlvEkQen4AyUgAOuQB3cggZoAgIm4Am8gFfr0Xq23qz3+WjOWuwcgz+wPr4BXceTpQ==</latexit><latexit sha1_base64="8yflH6XmxeFIYfP+0DsvkgYfluk=">AAACAHicbVDLSgMxFM3UV62vURcu3ASLUBctmSLYjVBw47KCfUA7Dpk004ZmMkOSKQxDN/6KGxeKuPUz3Pk3pg9EWw9c7uGce0nu8WPOlEboy8qtrW9sbuW3Czu7e/sH9uFRS0WJJLRJIh7Jjo8V5UzQpmaa004sKQ59Ttv+6Gbqt8dUKhaJe53G1A3xQLCAEayN5Nkn44cqvIZjD5lehtVBKS2nHrrw7CKqoBngD3GWSREs0PDsz14/IklIhSYcK9V1UKzdDEvNCKeTQi9RNMZkhAe0a6jAIVVuNjtgAs+N0odBJE0JDWfq740Mh0qloW8mQ6yHatmbiv953UQHNTdjIk40FWT+UJBwqCM4TQP2maRE89QQTCQzf4VkiCUm2mRWMCGsnLxKWtWKgyrO3WWxXlvEkQen4AyUgAOuQB3cggZoAgIm4Am8gFfr0Xq23qz3+WjOWuwcgz+wPr4BXceTpQ==</latexit><latexit sha1_base64="8yflH6XmxeFIYfP+0DsvkgYfluk=">AAACAHicbVDLSgMxFM3UV62vURcu3ASLUBctmSLYjVBw47KCfUA7Dpk004ZmMkOSKQxDN/6KGxeKuPUz3Pk3pg9EWw9c7uGce0nu8WPOlEboy8qtrW9sbuW3Czu7e/sH9uFRS0WJJLRJIh7Jjo8V5UzQpmaa004sKQ59Ttv+6Gbqt8dUKhaJe53G1A3xQLCAEayN5Nkn44cqvIZjD5lehtVBKS2nHrrw7CKqoBngD3GWSREs0PDsz14/IklIhSYcK9V1UKzdDEvNCKeTQi9RNMZkhAe0a6jAIVVuNjtgAs+N0odBJE0JDWfq740Mh0qloW8mQ6yHatmbiv953UQHNTdjIk40FWT+UJBwqCM4TQP2maRE89QQTCQzf4VkiCUm2mRWMCGsnLxKWtWKgyrO3WWxXlvEkQen4AyUgAOuQB3cggZoAgIm4Am8gFfr0Xq23qz3+WjOWuwcgz+wPr4BXceTpQ==</latexit><latexit sha1_base64="8yflH6XmxeFIYfP+0DsvkgYfluk=">AAACAHicbVDLSgMxFM3UV62vURcu3ASLUBctmSLYjVBw47KCfUA7Dpk004ZmMkOSKQxDN/6KGxeKuPUz3Pk3pg9EWw9c7uGce0nu8WPOlEboy8qtrW9sbuW3Czu7e/sH9uFRS0WJJLRJIh7Jjo8V5UzQpmaa004sKQ59Ttv+6Gbqt8dUKhaJe53G1A3xQLCAEayN5Nkn44cqvIZjD5lehtVBKS2nHrrw7CKqoBngD3GWSREs0PDsz14/IklIhSYcK9V1UKzdDEvNCKeTQi9RNMZkhAe0a6jAIVVuNjtgAs+N0odBJE0JDWfq740Mh0qloW8mQ6yHatmbiv953UQHNTdjIk40FWT+UJBwqCM4TQP2maRE89QQTCQzf4VkiCUm2mRWMCGsnLxKWtWKgyrO3WWxXlvEkQen4AyUgAOuQB3cggZoAgIm4Am8gFfr0Xq23qz3+WjOWuwcgz+wPr4BXceTpQ==</latexit>

=> v2 = v20<latexit sha1_base64="e7jiIZV4l+fmUbtCXxPx6gBNZ3c=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKewGwVwiAS8eI5gHJJswO5lNhszOLDOzkbDkP7x4UMSr/+LNv3HyQDSxoKGo6qa7K4g508Z1v5zMxubW9k52N7e3f3B4lD8+aWiZKELrRHKpWgHWlDNB64YZTluxojgKOG0Go9uZ3xxTpZkUD2YSUz/CA8FCRrCxUrdyg8bdEqqgcc/tlnr5glt050A/xFslBVii1st/dvqSJBEVhnCsddtzY+OnWBlGOJ3mOommMSYjPKBtSwWOqPbT+dVTdGGVPgqlsiUMmqu/J1IcaT2JAtsZYTPUq95M/M9rJyYs+ykTcWKoIItFYcKRkWgWAeozRYnhE0swUczeisgQK0yMDSpnQ1h7eZ00SkXPLXr3V4VqeRlHFs7gHC7Bg2uowh3UoA4EFDzBC7w6j86z8+a8L1ozznLmFP7A+fgGwiaQsA==</latexit><latexit sha1_base64="e7jiIZV4l+fmUbtCXxPx6gBNZ3c=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKewGwVwiAS8eI5gHJJswO5lNhszOLDOzkbDkP7x4UMSr/+LNv3HyQDSxoKGo6qa7K4g508Z1v5zMxubW9k52N7e3f3B4lD8+aWiZKELrRHKpWgHWlDNB64YZTluxojgKOG0Go9uZ3xxTpZkUD2YSUz/CA8FCRrCxUrdyg8bdEqqgcc/tlnr5glt050A/xFslBVii1st/dvqSJBEVhnCsddtzY+OnWBlGOJ3mOommMSYjPKBtSwWOqPbT+dVTdGGVPgqlsiUMmqu/J1IcaT2JAtsZYTPUq95M/M9rJyYs+ykTcWKoIItFYcKRkWgWAeozRYnhE0swUczeisgQK0yMDSpnQ1h7eZ00SkXPLXr3V4VqeRlHFs7gHC7Bg2uowh3UoA4EFDzBC7w6j86z8+a8L1ozznLmFP7A+fgGwiaQsA==</latexit><latexit sha1_base64="e7jiIZV4l+fmUbtCXxPx6gBNZ3c=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKewGwVwiAS8eI5gHJJswO5lNhszOLDOzkbDkP7x4UMSr/+LNv3HyQDSxoKGo6qa7K4g508Z1v5zMxubW9k52N7e3f3B4lD8+aWiZKELrRHKpWgHWlDNB64YZTluxojgKOG0Go9uZ3xxTpZkUD2YSUz/CA8FCRrCxUrdyg8bdEqqgcc/tlnr5glt050A/xFslBVii1st/dvqSJBEVhnCsddtzY+OnWBlGOJ3mOommMSYjPKBtSwWOqPbT+dVTdGGVPgqlsiUMmqu/J1IcaT2JAtsZYTPUq95M/M9rJyYs+ykTcWKoIItFYcKRkWgWAeozRYnhE0swUczeisgQK0yMDSpnQ1h7eZ00SkXPLXr3V4VqeRlHFs7gHC7Bg2uowh3UoA4EFDzBC7w6j86z8+a8L1ozznLmFP7A+fgGwiaQsA==</latexit><latexit sha1_base64="e7jiIZV4l+fmUbtCXxPx6gBNZ3c=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKewGwVwiAS8eI5gHJJswO5lNhszOLDOzkbDkP7x4UMSr/+LNv3HyQDSxoKGo6qa7K4g508Z1v5zMxubW9k52N7e3f3B4lD8+aWiZKELrRHKpWgHWlDNB64YZTluxojgKOG0Go9uZ3xxTpZkUD2YSUz/CA8FCRrCxUrdyg8bdEqqgcc/tlnr5glt050A/xFslBVii1st/dvqSJBEVhnCsddtzY+OnWBlGOJ3mOommMSYjPKBtSwWOqPbT+dVTdGGVPgqlsiUMmqu/J1IcaT2JAtsZYTPUq95M/M9rJyYs+ykTcWKoIItFYcKRkWgWAeozRYnhE0swUczeisgQK0yMDSpnQ1h7eZ00SkXPLXr3V4VqeRlHFs7gHC7Bg2uowh3UoA4EFDzBC7w6j86z8+a8L1ozznLmFP7A+fgGwiaQsA==</latexit>

=> v = 24.5m/s or � 24.5m/s<latexit sha1_base64="3Am9kpAgBbPbMaBoyiTrddNRFMg=">AAACCnicbVDLSgMxFM3UV62vUZduokVwUceZUrGbSsGNywr2AZ2hZNJMG5rMDEmmUIau3fgrblwo4tYvcOffmD4QbT1wL4dz7iW5x48Zlcq2v4zMyura+kZ2M7e1vbO7Z+4fNGSUCEzqOGKRaPlIEkZDUldUMdKKBUHcZ6TpD24mfnNIhKRReK9GMfE46oU0oBgpLXXM48o1HMIKLJasS34hoVtwCzASup3PpY6Zty17CvhDnEWSB3PUOuan241wwkmoMENSth07Vl6KhKKYkXHOTSSJER6gHmlrGiJOpJdOTxnDU610YRAJXaGCU/X3Roq4lCPu60mOVF8uehPxP6+dqKDspTSME0VCPHsoSBhUEZzkArtUEKzYSBOEBdV/hbiPBMJKp5fTISydvEwaRcuxLeeulK+W53FkwRE4AWfAAVegCm5BDdQBBg/gCbyAV+PReDbejPfZaMaY7xyCPzA+vgFQvpYw</latexit><latexit sha1_base64="3Am9kpAgBbPbMaBoyiTrddNRFMg=">AAACCnicbVDLSgMxFM3UV62vUZduokVwUceZUrGbSsGNywr2AZ2hZNJMG5rMDEmmUIau3fgrblwo4tYvcOffmD4QbT1wL4dz7iW5x48Zlcq2v4zMyura+kZ2M7e1vbO7Z+4fNGSUCEzqOGKRaPlIEkZDUldUMdKKBUHcZ6TpD24mfnNIhKRReK9GMfE46oU0oBgpLXXM48o1HMIKLJasS34hoVtwCzASup3PpY6Zty17CvhDnEWSB3PUOuan241wwkmoMENSth07Vl6KhKKYkXHOTSSJER6gHmlrGiJOpJdOTxnDU610YRAJXaGCU/X3Roq4lCPu60mOVF8uehPxP6+dqKDspTSME0VCPHsoSBhUEZzkArtUEKzYSBOEBdV/hbiPBMJKp5fTISydvEwaRcuxLeeulK+W53FkwRE4AWfAAVegCm5BDdQBBg/gCbyAV+PReDbejPfZaMaY7xyCPzA+vgFQvpYw</latexit><latexit sha1_base64="3Am9kpAgBbPbMaBoyiTrddNRFMg=">AAACCnicbVDLSgMxFM3UV62vUZduokVwUceZUrGbSsGNywr2AZ2hZNJMG5rMDEmmUIau3fgrblwo4tYvcOffmD4QbT1wL4dz7iW5x48Zlcq2v4zMyura+kZ2M7e1vbO7Z+4fNGSUCEzqOGKRaPlIEkZDUldUMdKKBUHcZ6TpD24mfnNIhKRReK9GMfE46oU0oBgpLXXM48o1HMIKLJasS34hoVtwCzASup3PpY6Zty17CvhDnEWSB3PUOuan241wwkmoMENSth07Vl6KhKKYkXHOTSSJER6gHmlrGiJOpJdOTxnDU610YRAJXaGCU/X3Roq4lCPu60mOVF8uehPxP6+dqKDspTSME0VCPHsoSBhUEZzkArtUEKzYSBOEBdV/hbiPBMJKp5fTISydvEwaRcuxLeeulK+W53FkwRE4AWfAAVegCm5BDdQBBg/gCbyAV+PReDbejPfZaMaY7xyCPzA+vgFQvpYw</latexit><latexit sha1_base64="3Am9kpAgBbPbMaBoyiTrddNRFMg=">AAACCnicbVDLSgMxFM3UV62vUZduokVwUceZUrGbSsGNywr2AZ2hZNJMG5rMDEmmUIau3fgrblwo4tYvcOffmD4QbT1wL4dz7iW5x48Zlcq2v4zMyura+kZ2M7e1vbO7Z+4fNGSUCEzqOGKRaPlIEkZDUldUMdKKBUHcZ6TpD24mfnNIhKRReK9GMfE46oU0oBgpLXXM48o1HMIKLJasS34hoVtwCzASup3PpY6Zty17CvhDnEWSB3PUOuan241wwkmoMENSth07Vl6KhKKYkXHOTSSJER6gHmlrGiJOpJdOTxnDU610YRAJXaGCU/X3Roq4lCPu60mOVF8uehPxP6+dqKDspTSME0VCPHsoSBhUEZzkArtUEKzYSBOEBdV/hbiPBMJKp5fTISydvEwaRcuxLeeulK+W53FkwRE4AWfAAVegCm5BDdQBBg/gCbyAV+PReDbejPfZaMaY7xyCPzA+vgFQvpYw</latexit>

A. Maller

Variable Acceleration

The examples we have done have all been for constant acceleration. With only algebra that is about all one can do.

However, it is straightforward to solve any acceleration problem with calculus.

The key thing is that you can’t use the equations of motion that we have been using, which are only true for constant acceleration. You have to solve the problem using calculus, just like we did to get the equations of motion for the special case of constant acceleration.

A. Maller

Example 3.17Motion of a Motorboat A motorboat is traveling at a constant velocity of 5.0 m/s when it starts to decelerate to arrive at the dock. Its acceleration is a(t)=−1/4t m/s3. (a) What is the velocity function of the motorboat? (b) At what time does the velocity reach zero? (c) What is the position function of the motorboat? (d) What is the displacement of the motorboat from the time it begins to decelerate to when the velocity is zero? (e) Graph the velocity and position functions.

dt<latexit sha1_base64="CGR5PD2HY71ZGFBNjnU68HhLP3I=">AAAB+nicdVDLSgMxFM34rPU11aWbYBFclaSIbRdC0Y3LCvYB7VAymUwbmnmQZCplnE9x40IRt36JO//GTFtBRQ8EDuecy705biy40gh9WCura+sbm4Wt4vbO7t6+XTroqCiRlLVpJCLZc4ligoesrbkWrBdLRgJXsK47ucr97pRJxaPwVs9i5gRkFHKfU6KNNLRLBF7A1JsOIpNKPZ1lQ7uMKgghjDHMCa6dI0MajXoV1yHOLYMyWKI1tN8HXkSTgIWaCqJUH6NYOymRmlPBsuIgUSwmdEJGrG9oSAKmnHR+egZPjOJBP5LmhRrO1e8TKQmUmgWuSQZEj9VvLxf/8vqJ9utOysM40Syki0V+IqCOYN4D9LhkVIuZIYRKbm6FdEwkodq0VTQlfP0U/k861QpGFXxzVm5eLusogCNwDE4BBjXQBNegBdqAgjvwAJ7As3VvPVov1usiumItZw7BD1hvn3IqlCM=</latexit><latexit sha1_base64="CGR5PD2HY71ZGFBNjnU68HhLP3I=">AAAB+nicdVDLSgMxFM34rPU11aWbYBFclaSIbRdC0Y3LCvYB7VAymUwbmnmQZCplnE9x40IRt36JO//GTFtBRQ8EDuecy705biy40gh9WCura+sbm4Wt4vbO7t6+XTroqCiRlLVpJCLZc4ligoesrbkWrBdLRgJXsK47ucr97pRJxaPwVs9i5gRkFHKfU6KNNLRLBF7A1JsOIpNKPZ1lQ7uMKgghjDHMCa6dI0MajXoV1yHOLYMyWKI1tN8HXkSTgIWaCqJUH6NYOymRmlPBsuIgUSwmdEJGrG9oSAKmnHR+egZPjOJBP5LmhRrO1e8TKQmUmgWuSQZEj9VvLxf/8vqJ9utOysM40Syki0V+IqCOYN4D9LhkVIuZIYRKbm6FdEwkodq0VTQlfP0U/k861QpGFXxzVm5eLusogCNwDE4BBjXQBNegBdqAgjvwAJ7As3VvPVov1usiumItZw7BD1hvn3IqlCM=</latexit><latexit sha1_base64="CGR5PD2HY71ZGFBNjnU68HhLP3I=">AAAB+nicdVDLSgMxFM34rPU11aWbYBFclaSIbRdC0Y3LCvYB7VAymUwbmnmQZCplnE9x40IRt36JO//GTFtBRQ8EDuecy705biy40gh9WCura+sbm4Wt4vbO7t6+XTroqCiRlLVpJCLZc4ligoesrbkWrBdLRgJXsK47ucr97pRJxaPwVs9i5gRkFHKfU6KNNLRLBF7A1JsOIpNKPZ1lQ7uMKgghjDHMCa6dI0MajXoV1yHOLYMyWKI1tN8HXkSTgIWaCqJUH6NYOymRmlPBsuIgUSwmdEJGrG9oSAKmnHR+egZPjOJBP5LmhRrO1e8TKQmUmgWuSQZEj9VvLxf/8vqJ9utOysM40Syki0V+IqCOYN4D9LhkVIuZIYRKbm6FdEwkodq0VTQlfP0U/k861QpGFXxzVm5eLusogCNwDE4BBjXQBNegBdqAgjvwAJ7As3VvPVov1usiumItZw7BD1hvn3IqlCM=</latexit><latexit sha1_base64="CGR5PD2HY71ZGFBNjnU68HhLP3I=">AAAB+nicdVDLSgMxFM34rPU11aWbYBFclaSIbRdC0Y3LCvYB7VAymUwbmnmQZCplnE9x40IRt36JO//GTFtBRQ8EDuecy705biy40gh9WCura+sbm4Wt4vbO7t6+XTroqCiRlLVpJCLZc4ligoesrbkWrBdLRgJXsK47ucr97pRJxaPwVs9i5gRkFHKfU6KNNLRLBF7A1JsOIpNKPZ1lQ7uMKgghjDHMCa6dI0MajXoV1yHOLYMyWKI1tN8HXkSTgIWaCqJUH6NYOymRmlPBsuIgUSwmdEJGrG9oSAKmnHR+egZPjOJBP5LmhRrO1e8TKQmUmgWuSQZEj9VvLxf/8vqJ9utOysM40Syki0V+IqCOYN4D9LhkVIuZIYRKbm6FdEwkodq0VTQlfP0U/k861QpGFXxzVm5eLusogCNwDE4BBjXQBNegBdqAgjvwAJ7As3VvPVov1usiumItZw7BD1hvn3IqlCM=</latexit>

=> v =

<latexit sha1_base64="SSxDOWigb2N5QN+sBFFgiWz2E6w=">AAAB+XicdVDLSgMxFM3UV62vUZdugkVwVZIitl1Uim5cVrAPaIeSyaRtaCYzJJlCGfonblwo4tY/ceffmGkrqOiBwOGce7knx48F1wahDye3tr6xuZXfLuzs7u0fuIdHbR0lirIWjUSkuj7RTHDJWoYbwbqxYiT0Bev4k5vM70yZ0jyS92YWMy8kI8mHnBJjpYHr1q/gFNZhn0sDCQzMwC2iEkIIYwwzgiuXyJJarVrGVYgzy6IIVmgO3Pd+ENEkZNJQQbTuYRQbLyXKcCrYvNBPNIsJnZAR61kqSci0ly6Sz+GZVQI4jJR9NsBC/b6RklDrWejbyZCYsf7tZeJfXi8xw6qXchknhkm6PDRMBDQRzGqAAVeMGjGzhFDFbVZIx0QRamxZBVvC10/h/6RdLmFUwncXxcb1qo48OAGn4BxgUAENcAuaoAUomIIH8ASendR5dF6c1+VozlntHIMfcN4+AVd+ki8=</latexit><latexit sha1_base64="SSxDOWigb2N5QN+sBFFgiWz2E6w=">AAAB+XicdVDLSgMxFM3UV62vUZdugkVwVZIitl1Uim5cVrAPaIeSyaRtaCYzJJlCGfonblwo4tY/ceffmGkrqOiBwOGce7knx48F1wahDye3tr6xuZXfLuzs7u0fuIdHbR0lirIWjUSkuj7RTHDJWoYbwbqxYiT0Bev4k5vM70yZ0jyS92YWMy8kI8mHnBJjpYHr1q/gFNZhn0sDCQzMwC2iEkIIYwwzgiuXyJJarVrGVYgzy6IIVmgO3Pd+ENEkZNJQQbTuYRQbLyXKcCrYvNBPNIsJnZAR61kqSci0ly6Sz+GZVQI4jJR9NsBC/b6RklDrWejbyZCYsf7tZeJfXi8xw6qXchknhkm6PDRMBDQRzGqAAVeMGjGzhFDFbVZIx0QRamxZBVvC10/h/6RdLmFUwncXxcb1qo48OAGn4BxgUAENcAuaoAUomIIH8ASendR5dF6c1+VozlntHIMfcN4+AVd+ki8=</latexit><latexit sha1_base64="SSxDOWigb2N5QN+sBFFgiWz2E6w=">AAAB+XicdVDLSgMxFM3UV62vUZdugkVwVZIitl1Uim5cVrAPaIeSyaRtaCYzJJlCGfonblwo4tY/ceffmGkrqOiBwOGce7knx48F1wahDye3tr6xuZXfLuzs7u0fuIdHbR0lirIWjUSkuj7RTHDJWoYbwbqxYiT0Bev4k5vM70yZ0jyS92YWMy8kI8mHnBJjpYHr1q/gFNZhn0sDCQzMwC2iEkIIYwwzgiuXyJJarVrGVYgzy6IIVmgO3Pd+ENEkZNJQQbTuYRQbLyXKcCrYvNBPNIsJnZAR61kqSci0ly6Sz+GZVQI4jJR9NsBC/b6RklDrWejbyZCYsf7tZeJfXi8xw6qXchknhkm6PDRMBDQRzGqAAVeMGjGzhFDFbVZIx0QRamxZBVvC10/h/6RdLmFUwncXxcb1qo48OAGn4BxgUAENcAuaoAUomIIH8ASendR5dF6c1+VozlntHIMfcN4+AVd+ki8=</latexit><latexit sha1_base64="SSxDOWigb2N5QN+sBFFgiWz2E6w=">AAAB+XicdVDLSgMxFM3UV62vUZdugkVwVZIitl1Uim5cVrAPaIeSyaRtaCYzJJlCGfonblwo4tY/ceffmGkrqOiBwOGce7knx48F1wahDye3tr6xuZXfLuzs7u0fuIdHbR0lirIWjUSkuj7RTHDJWoYbwbqxYiT0Bev4k5vM70yZ0jyS92YWMy8kI8mHnBJjpYHr1q/gFNZhn0sDCQzMwC2iEkIIYwwzgiuXyJJarVrGVYgzy6IIVmgO3Pd+ENEkZNJQQbTuYRQbLyXKcCrYvNBPNIsJnZAR61kqSci0ly6Sz+GZVQI4jJR9NsBC/b6RklDrWejbyZCYsf7tZeJfXi8xw6qXchknhkm6PDRMBDQRzGqAAVeMGjGzhFDFbVZIx0QRamxZBVvC10/h/6RdLmFUwncXxcb1qo48OAGn4BxgUAENcAuaoAUomIIH8ASendR5dF6c1+VozlntHIMfcN4+AVd+ki8=</latexit>

= 5.0� 1

<latexit sha1_base64="hoVfxY7BIHCOJ5Q1nJJCdnSRx94=">AAAB/3icdVDLSgMxFM3UV62vquDGTbAIbhySona6EApuXFawD2jHkkkzbWjmQZIRytiFv+LGhSJu/Q13/o2ZtoKKHrhwOOde7r3HiwVXGqEPK7ewuLS8kl8trK1vbG4Vt3eaKkokZQ0aiUi2PaKY4CFraK4Fa8eSkcATrOWNLjK/dcuk4lF4rccxcwMyCLnPKdFG6hX34Dk8tRE87vqS0BRPUmeib8q9YgnZCCGMMcwIrpwhQ6pVp4wdiDPLoATmqPeK791+RJOAhZoKolQHo1i7KZGaU8EmhW6iWEzoiAxYx9CQBEy56fT+CTw0Sh/6kTQVajhVv0+kJFBqHHimMyB6qH57mfiX10m077gpD+NEs5DOFvmJgDqCWRiwzyWjWowNIVRycyukQ2KC0Cayggnh61P4P2mWbYxsfHVSqjnzOPJgHxyAI4BBBdTAJaiDBqDgDjyAJ/Bs3VuP1ov1OmvNWfOZXfAD1tsnXW+UZQ==</latexit><latexit sha1_base64="hoVfxY7BIHCOJ5Q1nJJCdnSRx94=">AAAB/3icdVDLSgMxFM3UV62vquDGTbAIbhySona6EApuXFawD2jHkkkzbWjmQZIRytiFv+LGhSJu/Q13/o2ZtoKKHrhwOOde7r3HiwVXGqEPK7ewuLS8kl8trK1vbG4Vt3eaKkokZQ0aiUi2PaKY4CFraK4Fa8eSkcATrOWNLjK/dcuk4lF4rccxcwMyCLnPKdFG6hX34Dk8tRE87vqS0BRPUmeib8q9YgnZCCGMMcwIrpwhQ6pVp4wdiDPLoATmqPeK791+RJOAhZoKolQHo1i7KZGaU8EmhW6iWEzoiAxYx9CQBEy56fT+CTw0Sh/6kTQVajhVv0+kJFBqHHimMyB6qH57mfiX10m077gpD+NEs5DOFvmJgDqCWRiwzyWjWowNIVRycyukQ2KC0Cayggnh61P4P2mWbYxsfHVSqjnzOPJgHxyAI4BBBdTAJaiDBqDgDjyAJ/Bs3VuP1ov1OmvNWfOZXfAD1tsnXW+UZQ==</latexit><latexit sha1_base64="hoVfxY7BIHCOJ5Q1nJJCdnSRx94=">AAAB/3icdVDLSgMxFM3UV62vquDGTbAIbhySona6EApuXFawD2jHkkkzbWjmQZIRytiFv+LGhSJu/Q13/o2ZtoKKHrhwOOde7r3HiwVXGqEPK7ewuLS8kl8trK1vbG4Vt3eaKkokZQ0aiUi2PaKY4CFraK4Fa8eSkcATrOWNLjK/dcuk4lF4rccxcwMyCLnPKdFG6hX34Dk8tRE87vqS0BRPUmeib8q9YgnZCCGMMcwIrpwhQ6pVp4wdiDPLoATmqPeK791+RJOAhZoKolQHo1i7KZGaU8EmhW6iWEzoiAxYx9CQBEy56fT+CTw0Sh/6kTQVajhVv0+kJFBqHHimMyB6qH57mfiX10m077gpD+NEs5DOFvmJgDqCWRiwzyWjWowNIVRycyukQ2KC0Cayggnh61P4P2mWbYxsfHVSqjnzOPJgHxyAI4BBBdTAJaiDBqDgDjyAJ/Bs3VuP1ov1OmvNWfOZXfAD1tsnXW+UZQ==</latexit><latexit sha1_base64="hoVfxY7BIHCOJ5Q1nJJCdnSRx94=">AAAB/3icdVDLSgMxFM3UV62vquDGTbAIbhySona6EApuXFawD2jHkkkzbWjmQZIRytiFv+LGhSJu/Q13/o2ZtoKKHrhwOOde7r3HiwVXGqEPK7ewuLS8kl8trK1vbG4Vt3eaKkokZQ0aiUi2PaKY4CFraK4Fa8eSkcATrOWNLjK/dcuk4lF4rccxcwMyCLnPKdFG6hX34Dk8tRE87vqS0BRPUmeib8q9YgnZCCGMMcwIrpwhQ6pVp4wdiDPLoATmqPeK791+RJOAhZoKolQHo1i7KZGaU8EmhW6iWEzoiAxYx9CQBEy56fT+CTw0Sh/6kTQVajhVv0+kJFBqHHimMyB6qH57mfiX10m077gpD+NEs5DOFvmJgDqCWRiwzyWjWowNIVRycyukQ2KC0Cayggnh61P4P2mWbYxsfHVSqjnzOPJgHxyAI4BBBdTAJaiDBqDgDjyAJ/Bs3VuP1ov1OmvNWfOZXfAD1tsnXW+UZQ==</latexit>

=> v =

4tdt = �1

8t2 + C1

<latexit sha1_base64="zeK/8zDITivQnvjHhTSvJY0aoP0=">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</latexit><latexit sha1_base64="zeK/8zDITivQnvjHhTSvJY0aoP0=">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</latexit><latexit sha1_base64="zeK/8zDITivQnvjHhTSvJY0aoP0=">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</latexit><latexit sha1_base64="zeK/8zDITivQnvjHhTSvJY0aoP0=">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</latexit>

v = 5.0� 1

8t2 = 0.0

<latexit sha1_base64="+Q2xOaT5sliHIs+J1iJxTr9bTp0=">AAACBnicdVDLSgMxFM34rPVVdSlCsAhuHJKidroQCm5cVrAPaMeSSTNtaOZBkimUoSs3/oobF4q49Rvc+Tdm2goqeiDh5Jx7ubnHiwVXGqEPa2FxaXllNbeWX9/Y3Nou7Ow2VJRIyuo0EpFseUQxwUNW11wL1oolI4EnWNMbXmZ+c8Sk4lF4o8cxcwPSD7nPKdFG6hYORvACntkInnR8SWiKJ6kz0bcloyIbdQtFcyOEMc6eCJfPkSGVilPCDsSZZVAEc9S6hfdOL6JJwEJNBVGqjVGs3ZRIzalgk3wnUSwmdEj6rG1oSAKm3HS6xgQeGaUH/UiaE2o4Vb93pCRQahx4pjIgeqB+e5n4l9dOtO+4KQ/jRLOQzgb5iYA6glkmsMclo1qMDSFUcvNXSAfExKFNcnkTwtem8H/SKNkY2fj6tFh15nHkwD44BMcAgzKogitQA3VAwR14AE/g2bq3Hq0X63VWumDNe/bAD1hvn+Ebliw=</latexit><latexit sha1_base64="+Q2xOaT5sliHIs+J1iJxTr9bTp0=">AAACBnicdVDLSgMxFM34rPVVdSlCsAhuHJKidroQCm5cVrAPaMeSSTNtaOZBkimUoSs3/oobF4q49Rvc+Tdm2goqeiDh5Jx7ubnHiwVXGqEPa2FxaXllNbeWX9/Y3Nou7Ow2VJRIyuo0EpFseUQxwUNW11wL1oolI4EnWNMbXmZ+c8Sk4lF4o8cxcwPSD7nPKdFG6hYORvACntkInnR8SWiKJ6kz0bcloyIbdQtFcyOEMc6eCJfPkSGVilPCDsSZZVAEc9S6hfdOL6JJwEJNBVGqjVGs3ZRIzalgk3wnUSwmdEj6rG1oSAKm3HS6xgQeGaUH/UiaE2o4Vb93pCRQahx4pjIgeqB+e5n4l9dOtO+4KQ/jRLOQzgb5iYA6glkmsMclo1qMDSFUcvNXSAfExKFNcnkTwtem8H/SKNkY2fj6tFh15nHkwD44BMcAgzKogitQA3VAwR14AE/g2bq3Hq0X63VWumDNe/bAD1hvn+Ebliw=</latexit><latexit sha1_base64="+Q2xOaT5sliHIs+J1iJxTr9bTp0=">AAACBnicdVDLSgMxFM34rPVVdSlCsAhuHJKidroQCm5cVrAPaMeSSTNtaOZBkimUoSs3/oobF4q49Rvc+Tdm2goqeiDh5Jx7ubnHiwVXGqEPa2FxaXllNbeWX9/Y3Nou7Ow2VJRIyuo0EpFseUQxwUNW11wL1oolI4EnWNMbXmZ+c8Sk4lF4o8cxcwPSD7nPKdFG6hYORvACntkInnR8SWiKJ6kz0bcloyIbdQtFcyOEMc6eCJfPkSGVilPCDsSZZVAEc9S6hfdOL6JJwEJNBVGqjVGs3ZRIzalgk3wnUSwmdEj6rG1oSAKm3HS6xgQeGaUH/UiaE2o4Vb93pCRQahx4pjIgeqB+e5n4l9dOtO+4KQ/jRLOQzgb5iYA6glkmsMclo1qMDSFUcvNXSAfExKFNcnkTwtem8H/SKNkY2fj6tFh15nHkwD44BMcAgzKogitQA3VAwR14AE/g2bq3Hq0X63VWumDNe/bAD1hvn+Ebliw=</latexit><latexit sha1_base64="+Q2xOaT5sliHIs+J1iJxTr9bTp0=">AAACBnicdVDLSgMxFM34rPVVdSlCsAhuHJKidroQCm5cVrAPaMeSSTNtaOZBkimUoSs3/oobF4q49Rvc+Tdm2goqeiDh5Jx7ubnHiwVXGqEPa2FxaXllNbeWX9/Y3Nou7Ow2VJRIyuo0EpFseUQxwUNW11wL1oolI4EnWNMbXmZ+c8Sk4lF4o8cxcwPSD7nPKdFG6hYORvACntkInnR8SWiKJ6kz0bcloyIbdQtFcyOEMc6eCJfPkSGVilPCDsSZZVAEc9S6hfdOL6JJwEJNBVGqjVGs3ZRIzalgk3wnUSwmdEj6rG1oSAKm3HS6xgQeGaUH/UiaE2o4Vb93pCRQahx4pjIgeqB+e5n4l9dOtO+4KQ/jRLOQzgb5iYA6glkmsMclo1qMDSFUcvNXSAfExKFNcnkTwtem8H/SKNkY2fj6tFh15nHkwD44BMcAgzKogitQA3VAwR14AE/g2bq3Hq0X63VWumDNe/bAD1hvn+Ebliw=</latexit>

=> t2 = 8⇥ 5.0 = 40<latexit sha1_base64="886wvhvmmZqoa2jeTsr0UdOtstA=">AAACA3icdVDLSgMxFM34rPU16k43wSK4KknxMV1UBDcuFawW2rFk0rSGZh4kd4QyFNz4K25cKOLWn3Dn35hpK6jogcC559zLzT1BoqQBQj6cqemZ2bn5wkJxcWl5ZdVdW780caq5qPNYxboRMCOUjEQdJCjRSLRgYaDEVdA/yf2rW6GNjKMLGCTCD1kvkl3JGVip7W7WjjBcV3ANe7gFMhQG75eJLfdI2y2RMiGEUopzQg8PiCXVqlehHqa5ZVFCE5y13fdWJ+ZpKCLgihnTpCQBP2MaJFdiWGylRiSM91lPNC2NmN3mZ6MbhnjHKh3cjbV9EeCR+n0iY6ExgzCwnSGDG/Pby8W/vGYKXc/PZJSkICI+XtRNFYYY54HgjtSCgxpYwriW9q+Y3zDNONjYijaEr0vx/+SyUqakTM/3SsfeJI4C2kLbaBdRdIiO0Sk6Q3XE0R16QE/o2bl3Hp0X53XcOuVMZjbQDzhvn8jolGE=</latexit><latexit sha1_base64="886wvhvmmZqoa2jeTsr0UdOtstA=">AAACA3icdVDLSgMxFM34rPU16k43wSK4KknxMV1UBDcuFawW2rFk0rSGZh4kd4QyFNz4K25cKOLWn3Dn35hpK6jogcC559zLzT1BoqQBQj6cqemZ2bn5wkJxcWl5ZdVdW780caq5qPNYxboRMCOUjEQdJCjRSLRgYaDEVdA/yf2rW6GNjKMLGCTCD1kvkl3JGVip7W7WjjBcV3ANe7gFMhQG75eJLfdI2y2RMiGEUopzQg8PiCXVqlehHqa5ZVFCE5y13fdWJ+ZpKCLgihnTpCQBP2MaJFdiWGylRiSM91lPNC2NmN3mZ6MbhnjHKh3cjbV9EeCR+n0iY6ExgzCwnSGDG/Pby8W/vGYKXc/PZJSkICI+XtRNFYYY54HgjtSCgxpYwriW9q+Y3zDNONjYijaEr0vx/+SyUqakTM/3SsfeJI4C2kLbaBdRdIiO0Sk6Q3XE0R16QE/o2bl3Hp0X53XcOuVMZjbQDzhvn8jolGE=</latexit><latexit sha1_base64="886wvhvmmZqoa2jeTsr0UdOtstA=">AAACA3icdVDLSgMxFM34rPU16k43wSK4KknxMV1UBDcuFawW2rFk0rSGZh4kd4QyFNz4K25cKOLWn3Dn35hpK6jogcC559zLzT1BoqQBQj6cqemZ2bn5wkJxcWl5ZdVdW780caq5qPNYxboRMCOUjEQdJCjRSLRgYaDEVdA/yf2rW6GNjKMLGCTCD1kvkl3JGVip7W7WjjBcV3ANe7gFMhQG75eJLfdI2y2RMiGEUopzQg8PiCXVqlehHqa5ZVFCE5y13fdWJ+ZpKCLgihnTpCQBP2MaJFdiWGylRiSM91lPNC2NmN3mZ6MbhnjHKh3cjbV9EeCR+n0iY6ExgzCwnSGDG/Pby8W/vGYKXc/PZJSkICI+XtRNFYYY54HgjtSCgxpYwriW9q+Y3zDNONjYijaEr0vx/+SyUqakTM/3SsfeJI4C2kLbaBdRdIiO0Sk6Q3XE0R16QE/o2bl3Hp0X53XcOuVMZjbQDzhvn8jolGE=</latexit><latexit sha1_base64="886wvhvmmZqoa2jeTsr0UdOtstA=">AAACA3icdVDLSgMxFM34rPU16k43wSK4KknxMV1UBDcuFawW2rFk0rSGZh4kd4QyFNz4K25cKOLWn3Dn35hpK6jogcC559zLzT1BoqQBQj6cqemZ2bn5wkJxcWl5ZdVdW780caq5qPNYxboRMCOUjEQdJCjRSLRgYaDEVdA/yf2rW6GNjKMLGCTCD1kvkl3JGVip7W7WjjBcV3ANe7gFMhQG75eJLfdI2y2RMiGEUopzQg8PiCXVqlehHqa5ZVFCE5y13fdWJ+ZpKCLgihnTpCQBP2MaJFdiWGylRiSM91lPNC2NmN3mZ6MbhnjHKh3cjbV9EeCR+n0iY6ExgzCwnSGDG/Pby8W/vGYKXc/PZJSkICI+XtRNFYYY54HgjtSCgxpYwriW9q+Y3zDNONjYijaEr0vx/+SyUqakTM/3SsfeJI4C2kLbaBdRdIiO0Sk6Q3XE0R16QE/o2bl3Hp0X53XcOuVMZjbQDzhvn8jolGE=</latexit>

=> t =p40 = 6.32s

<latexit sha1_base64="X+UqoP3vO/7bnF61si9r+5ylWeo=">AAACAnicdZDLSgMxFIYzXmu9VV2Jm2ARXJWkljouKoIblwq2FdqhZNLUhmYuJmeEMhQ3voobF4q49Snc+TZm2goqeiDw8f/ncHJ+P1bSACEfzszs3PzCYm4pv7yyurZe2NhsmCjRXNR5pCJ95TMjlAxFHSQocRVrwQJfiaY/OM385q3QRkbhJQxj4QXsOpQ9yRlYqVPYrh1jwDXcNjca0goZWa6WDsrYdApFUiKEUEpxBvSwSiwcHbll6mKaWbaKaFrnncJ7uxvxJBAhcMWMaVESg5cyDZIrMcq3EyNixgfsWrQshiwQxkvHJ4zwnlW6uBdp+0LAY/X7RMoCY4aBbzsDBn3z28vEv7xWAj3XS2UYJyBCPlnUSxSGCGd54K7UgoMaWmBcS/tXzPtMMw42tbwN4etS/D80yiVKSvSiUjxxp3Hk0A7aRfuIokN0gs7QOaojju7QA3pCz8698+i8OK+T1hlnOrOFfpTz9gk3MJS4</latexit><latexit sha1_base64="X+UqoP3vO/7bnF61si9r+5ylWeo=">AAACAnicdZDLSgMxFIYzXmu9VV2Jm2ARXJWkljouKoIblwq2FdqhZNLUhmYuJmeEMhQ3voobF4q49Snc+TZm2goqeiDw8f/ncHJ+P1bSACEfzszs3PzCYm4pv7yyurZe2NhsmCjRXNR5pCJ95TMjlAxFHSQocRVrwQJfiaY/OM385q3QRkbhJQxj4QXsOpQ9yRlYqVPYrh1jwDXcNjca0goZWa6WDsrYdApFUiKEUEpxBvSwSiwcHbll6mKaWbaKaFrnncJ7uxvxJBAhcMWMaVESg5cyDZIrMcq3EyNixgfsWrQshiwQxkvHJ4zwnlW6uBdp+0LAY/X7RMoCY4aBbzsDBn3z28vEv7xWAj3XS2UYJyBCPlnUSxSGCGd54K7UgoMaWmBcS/tXzPtMMw42tbwN4etS/D80yiVKSvSiUjxxp3Hk0A7aRfuIokN0gs7QOaojju7QA3pCz8698+i8OK+T1hlnOrOFfpTz9gk3MJS4</latexit><latexit sha1_base64="X+UqoP3vO/7bnF61si9r+5ylWeo=">AAACAnicdZDLSgMxFIYzXmu9VV2Jm2ARXJWkljouKoIblwq2FdqhZNLUhmYuJmeEMhQ3voobF4q49Snc+TZm2goqeiDw8f/ncHJ+P1bSACEfzszs3PzCYm4pv7yyurZe2NhsmCjRXNR5pCJ95TMjlAxFHSQocRVrwQJfiaY/OM385q3QRkbhJQxj4QXsOpQ9yRlYqVPYrh1jwDXcNjca0goZWa6WDsrYdApFUiKEUEpxBvSwSiwcHbll6mKaWbaKaFrnncJ7uxvxJBAhcMWMaVESg5cyDZIrMcq3EyNixgfsWrQshiwQxkvHJ4zwnlW6uBdp+0LAY/X7RMoCY4aBbzsDBn3z28vEv7xWAj3XS2UYJyBCPlnUSxSGCGd54K7UgoMaWmBcS/tXzPtMMw42tbwN4etS/D80yiVKSvSiUjxxp3Hk0A7aRfuIokN0gs7QOaojju7QA3pCz8698+i8OK+T1hlnOrOFfpTz9gk3MJS4</latexit><latexit sha1_base64="X+UqoP3vO/7bnF61si9r+5ylWeo=">AAACAnicdZDLSgMxFIYzXmu9VV2Jm2ARXJWkljouKoIblwq2FdqhZNLUhmYuJmeEMhQ3voobF4q49Snc+TZm2goqeiDw8f/ncHJ+P1bSACEfzszs3PzCYm4pv7yyurZe2NhsmCjRXNR5pCJ95TMjlAxFHSQocRVrwQJfiaY/OM385q3QRkbhJQxj4QXsOpQ9yRlYqVPYrh1jwDXcNjca0goZWa6WDsrYdApFUiKEUEpxBvSwSiwcHbll6mKaWbaKaFrnncJ7uxvxJBAhcMWMaVESg5cyDZIrMcq3EyNixgfsWrQshiwQxkvHJ4zwnlW6uBdp+0LAY/X7RMoCY4aBbzsDBn3z28vEv7xWAj3XS2UYJyBCPlnUSxSGCGd54K7UgoMaWmBcS/tXzPtMMw42tbwN4etS/D80yiVKSvSiUjxxp3Hk0A7aRfuIokN0gs7QOaojju7QA3pCz8698+i8OK+T1hlnOrOFfpTz9gk3MJS4</latexit>

dt<latexit sha1_base64="EV8L4Lo+DkmRTpZR4tmeyvcsdMk=">AAAB+nicdVDLSgMxFM34rPU11aWbYBFclaSInS6EghuXFewD2qFkMpk2NPMgyVTL2E9x40IRt36JO//GTFtBRQ8EDuecy705XiK40gh9WCura+sbm4Wt4vbO7t6+XTpoqziVlLVoLGLZ9YhigkespbkWrJtIRkJPsI43vsz9zoRJxePoRk8T5oZkGPGAU6KNNLBLE3gBM/+uH5tU5uvZbGCXUQUhhDGGOcG1c2RIve5UsQNxbhmUwRLNgf3e92OahizSVBClehgl2s2I1JwKNiv2U8USQsdkyHqGRiRkys3mp8/giVF8GMTSvEjDufp9IiOhUtPQM8mQ6JH67eXiX14v1YHjZjxKUs0iulgUpALqGOY9QJ9LRrWYGkKo5OZWSEdEEqpNW0VTwtdP4f+kXa1gVMHXZ+WGs6yjAI7AMTgFGNRAA1yBJmgBCm7BA3gCz9a99Wi9WK+L6Iq1nDkEP2C9fQKTaJQw</latexit><latexit sha1_base64="EV8L4Lo+DkmRTpZR4tmeyvcsdMk=">AAAB+nicdVDLSgMxFM34rPU11aWbYBFclaSInS6EghuXFewD2qFkMpk2NPMgyVTL2E9x40IRt36JO//GTFtBRQ8EDuecy705XiK40gh9WCura+sbm4Wt4vbO7t6+XTpoqziVlLVoLGLZ9YhigkespbkWrJtIRkJPsI43vsz9zoRJxePoRk8T5oZkGPGAU6KNNLBLE3gBM/+uH5tU5uvZbGCXUQUhhDGGOcG1c2RIve5UsQNxbhmUwRLNgf3e92OahizSVBClehgl2s2I1JwKNiv2U8USQsdkyHqGRiRkys3mp8/giVF8GMTSvEjDufp9IiOhUtPQM8mQ6JH67eXiX14v1YHjZjxKUs0iulgUpALqGOY9QJ9LRrWYGkKo5OZWSEdEEqpNW0VTwtdP4f+kXa1gVMHXZ+WGs6yjAI7AMTgFGNRAA1yBJmgBCm7BA3gCz9a99Wi9WK+L6Iq1nDkEP2C9fQKTaJQw</latexit><latexit sha1_base64="EV8L4Lo+DkmRTpZR4tmeyvcsdMk=">AAAB+nicdVDLSgMxFM34rPU11aWbYBFclaSInS6EghuXFewD2qFkMpk2NPMgyVTL2E9x40IRt36JO//GTFtBRQ8EDuecy705XiK40gh9WCura+sbm4Wt4vbO7t6+XTpoqziVlLVoLGLZ9YhigkespbkWrJtIRkJPsI43vsz9zoRJxePoRk8T5oZkGPGAU6KNNLBLE3gBM/+uH5tU5uvZbGCXUQUhhDGGOcG1c2RIve5UsQNxbhmUwRLNgf3e92OahizSVBClehgl2s2I1JwKNiv2U8USQsdkyHqGRiRkys3mp8/giVF8GMTSvEjDufp9IiOhUtPQM8mQ6JH67eXiX14v1YHjZjxKUs0iulgUpALqGOY9QJ9LRrWYGkKo5OZWSEdEEqpNW0VTwtdP4f+kXa1gVMHXZ+WGs6yjAI7AMTgFGNRAA1yBJmgBCm7BA3gCz9a99Wi9WK+L6Iq1nDkEP2C9fQKTaJQw</latexit><latexit sha1_base64="EV8L4Lo+DkmRTpZR4tmeyvcsdMk=">AAAB+nicdVDLSgMxFM34rPU11aWbYBFclaSInS6EghuXFewD2qFkMpk2NPMgyVTL2E9x40IRt36JO//GTFtBRQ8EDuecy705XiK40gh9WCura+sbm4Wt4vbO7t6+XTpoqziVlLVoLGLZ9YhigkespbkWrJtIRkJPsI43vsz9zoRJxePoRk8T5oZkGPGAU6KNNLBLE3gBM/+uH5tU5uvZbGCXUQUhhDGGOcG1c2RIve5UsQNxbhmUwRLNgf3e92OahizSVBClehgl2s2I1JwKNiv2U8USQsdkyHqGRiRkys3mp8/giVF8GMTSvEjDufp9IiOhUtPQM8mQ6JH67eXiX14v1YHjZjxKUs0iulgUpALqGOY9QJ9LRrWYGkKo5OZWSEdEEqpNW0VTwtdP4f+kXa1gVMHXZ+WGs6yjAI7AMTgFGNRAA1yBJmgBCm7BA3gCz9a99Wi9WK+L6Iq1nDkEP2C9fQKTaJQw</latexit>

=> x =

Zvdt =

Z(5.0� 1

8t2)dt

<latexit sha1_base64="QyLn/aoB3Ich2iIHSvdwwWOE+ME=">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</latexit><latexit sha1_base64="QyLn/aoB3Ich2iIHSvdwwWOE+ME=">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</latexit><latexit sha1_base64="QyLn/aoB3Ich2iIHSvdwwWOE+ME=">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</latexit><latexit sha1_base64="QyLn/aoB3Ich2iIHSvdwwWOE+ME=">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</latexit>

= 5.0t� 1

24t3 + C2

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= 5.0t� 1

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x(6.32s) = 50(6.32)� 6.323/24 = 21.1m<latexit sha1_base64="cl1Z+/hjMXt261qK3rf4rmhDxuY=">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</latexit><latexit sha1_base64="cl1Z+/hjMXt261qK3rf4rmhDxuY=">AAACEHicdZDLSgMxFIYzXmu9VV26CRZRF47JeGsXQsGNywpWhVpLJs20ocnMkGTEMvQR3Pgqblwo4talO9/GTFtBRQ+EfPz/OSTn92PBtUHowxkbn5icms7N5Gfn5hcWC0vL5zpKFGU1GolIXfpEM8FDVjPcCHYZK0akL9iF3z3O/IsbpjSPwjPTi1lDknbIA06JsVKzsHG7eeDuenoLHsF9NOAtuA2z+3p3x9uzsoddLJuFInIRQhhjmAE+PEAWyuWSh0sQZ5atIhhVtVl4v2pFNJEsNFQQresYxaaREmU4Fayfv0o0iwntkjarWwyJZLqRDhbqw3WrtGAQKXtCAwfq94mUSK170redkpiO/u1l4l9ePTFBqZHyME4MC+nwoSAR0EQwSwe2uGLUiJ4FQhW3f4W0QxShxmaYtyF8bQr/h3PPxcjFp3vFSmkURw6sgjWwCTA4BBVwAqqgBii4Aw/gCTw7986j8+K8DlvHnNHMCvhRztsnPiuWZg==</latexit><latexit sha1_base64="cl1Z+/hjMXt261qK3rf4rmhDxuY=">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</latexit><latexit sha1_base64="cl1Z+/hjMXt261qK3rf4rmhDxuY=">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</latexit>

A. Maller

Homework

Chapter 3 - 28,29,35,41,42,45,51,68

Kinematics - City Tech OpenLab

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