Physics 121

2. Motion in one Dimension

2.1 Reference Frames and Displacement

2.2 Average Velocity

2.3 Instantaneous Velocity

2.4 Acceleration

2.5 Motion at Constant Acceleration

2.6 How to Solve Problems

2.7 Falling Objects

2.8 Graphs of Linear Motion

Speed

Speed = Distance / Time

v = d / t

Example 2.1 . . . From a Distance

If you are driving 110 km/h along a straight road and you look to the side for 2.0 s, how far do you travel during this inattentive period?

Solution 2.1 . . . From a Distance

v = d/t30.6 m/s = d / 2.0 sd = 30.6 m/s x 2.0 sd = 61 m

Given: v = 110 km/h t = 2.0 s Solve for: d

110 km/h = 110 km/h x 1000 m/km / 3600 s/h

110 km/h = 30.6 m/s

Formula v = d/t

But first … convert speed (v) into m/s!!!

Distance and Displacement

Displacement measures the change in position of an object. Also, the direction, in addition to the magnitude, must be considered.

Distance and Displacement can be very different if the object does not proceed in the same direction in a straight line!

Example 2.2 . . . Around the block

A BC

A car travels East from point A to point B (5 miles) and then back (West) from point B to point C (2 miles).

(a) What distance did the car travel?(b) What is the car’s displacement?

Solution 2.2 . . . Around the block

A BC

A car travels East from point A to point B (5 miles) and then back (West) from point B to point C (2 miles).

(a) Distance traveled is 7 miles(b) Displacement is 3 miles East

Speed and Velocity

Velocity = displacement / time

•Car A going at 30 m.p.h. EAST and car B going at 30 m.p.h. NORTH have the same speed but different velocities!

•A car going around a circular track at 30 m.p.h. has a CONSTANT SPEED but its VELOCITY is CHANGING!

Example 2.3 . . . Speed and Velocity

John Denver negotiates his rusty (but trusty) truck around a bend in a country road in West Virginia.

A. His speed is constant but his velocity is changing

B. His velocity is constant but his speed is changing

C. Both his speed and velocity are changing

D. His velocity is definitely changing but his speed may or may not be changing

Solution 2.3 . . . Speed and Velocity

John Denver negotiates his rusty (but trusty) truck around a bend in a country road in West Virginia.

A. His speed is constant but his velocity is changing

B. His velocity is constant but his speed is changing

C. Both his speed and velocity are changing

D. His velocity is definitely changing but his speed may or may not be changing

Example 2.4 . . . On the road again!

A BC

A car travels East from point A to point B (5 miles) and then back (West) from point B to point C (2 miles) in 15 minutes

(a) What is the speed of the car?(b) What is velocity of the car?

Solution 2.4 . . . On the road again!

A BC

A car travels East from point A to point B (5 miles) and then back (West) from point B to point C (2 miles).

(a) Speed is 28 mph(b) Velocity is 12 mph East

Instantaneous Velocity

• Instantaneous means measured at a given instant or moment (“Kodak moment”). Experimentally, this is virtually impossible. One must measure the distance over an extremely short time interval

• Average means measured over an extended time interval. This is easier to measure.

Speed and Velocity . . . revisited!

• Speed = distance / time

• Velocity = displacement / time

• For translational motion (straight line) in the same direction, speed equals velocity!

• Speed is an example of a SCALAR quantity. Only the magnitude (amount) is specified without regard to the direction

• Remember to specify the magnitude and the direction for displacement and velocity. These are VECTOR quantities

Acceleration

Acceleration is the rate of change of velocity

a = ( vf - vi ) / t

Example 2.5 . . . Uniformly accelerating car

When the traffic light turns green, the speed of a car increases uniformly from vi = 0 m/s at t = 0 s to vf = 18 m/s at t =6 s

(a) Calculate the acceleration(b) Calculate the speed at t = 4 s(c) Calculate the distance traveled in 6 s

Solution 2.5 . . . Uniformly accelerating car

a = (Vf - Vi ) / ta = (18 - 0) / 6a = 3 m/s2

Vf = Vi + a tVf = 0 + 3x4Vf at 4s = 12 m/s

Vave = (Vf + Vi ) / 2

Vave = (18+0)/2

Vave = 9 m /s

Vave = d / t

9 m/s = d / 6 s

d = 54 m

Example 2.6 . . . Motion in one dimension

• Is the speed constant at t=10s?

• When is the acceleration zero?

• When is it slowing down (decelerating)?

• What is the acceleration at t=7s?

Solution 2.6 . . . Motion in one dimension

• Is the speed constant at t=10s? No

• When is the acceleration zero? 13 s < t < 42 s

• When is it slowing down (decelerating)? t > 42 s

• What is the acceleration at t=7s? 1 m / s2

Equation Summary for uniform acceleration (No Jerks Pleeeease!)

Vave = d / t

Vf = Vi + a t

Vave = (Vf + Vi) / 2

d = Vi t + 1/2 a t2

Vf2 = Vi

2 + 2ad

Example 2.7 . . . Driver’s not a jerk!

A car accelerates from a stop light and attains a speed of 24 m/s after traveling a distance of 72 m. What is the acceleration of the car?

Solution 2.7 . . . Driver’s not a jerk!

(24)2 = 0 + (2)(a)(72)a = 4 m/s2

Given: Vi = 0 Vf = 24 m/s d = 72 m Solve for: a

Formula Vf2 = Vi

2 + 2ad

Example 2.8 . . . Free Fall

Acceleration due to Earth’s gravity is 9.8 m/s2

(a) A cat falls off a ledge. How fast is it moving 3 seconds after the fall?

(b) What is the distance traveled by the unfortunate cat?

Solution 2.8 . . . Free Fall

Vf = 0 + (9.8)(3)Vf = 30 m/s

Given: Vi = 0 a = 9.8 m/s2 t = 3 s Solve for: Vf d

Formula Vf = Vi + a t d = Vi t + 1/2 a t2

d = 0 + (1/2)(9.8)(3)(3)d = 44 m

Example 2.9 . . . Granny’s Orchard

Big apple is twice as big as Little apple. Both fall from the same height at the same time. Which statement is correct?

A. Both reach the ground at about the same time

B. Big reaches the ground way before Little does

C. Little reaches the ground way before Big does

Solution 2.9 . . . Granny’s Orchard

Big apple is twice as big as Little apple. Both fall from the same height at the same time. Which statement is correct?

A. Both reach the ground at about the same time

B. Big reaches the ground way before Little does

C. Little reaches the ground way before Big does

That’s all folks!