1

Physics 1110: Motion in 1D

•  Wasn’t around yesterday; Had a 65 ft spruce tree crash

thru my roof in the wind storm

Good talk on Sat at 2pm in G1B30 by Professor Emeritus

Albert Bartlett

Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/

Announcements:

•  CAPA homework due Tuesday at 10pm

2

Vectors •  Vectors have magnitude and direction

– velocity, acceleration, displacement •  Scalars only have magnitude

– speed, temperature, mass, volume •  Vector quantities are indicated by bold and/or an

arrow above: •  Can use arrow to represent direction & magnitude

3

Reference frame

•  For 1D motion, we choose the x-axis to lie along the direction of motion so the y-axis is unnecessary

Standard 2D reference frame is motionless with respect to the Earth and has x and y axes as shown

•  To solve any problem, need to pick a reference frame and conduct all calculations in that frame

+y

-x -y

+x

4

1D kinematics (motion) •  Define two quantities related to change in position:

– Distance is a scalar which can be thought of as the ground covered by the chosen path

– Displacement is a vector between the start and end points whose magnitude is the distance between the start and end points:

•  In 1D, can simplify to . The vector direction can be along positive or negative axis so the signs of x2 and x1 are still relevant.

5

Displacement and Distance •  A person swims one lap (round-

trip) in an Olympic-sized pool (50m long). What is the distance traveled and the displacement?

0 50 100 -100 50 x

(m)

Distance = 100 m

Displacement = Δx = x2 - x1 = 0 - 0 = 0 m

50 100 -100 50 x

(m) 0

•  Place origin at the starting point – Outbound trip: – Return trip:

An ant crawling along the floor follows a semi-circular path, going half way around the circumference of a circle of radius R.

6

Clicker question 1 Set frequency to BA

Thedistancetraveled,andthedisplacementoftheant,arerespectively:

A: π R and π R

B: 2 R and π R

C: π R and 2 R

D: π R and zero

E: none of these

7

Clicker question 2 Set frequency to BA

An object goes from one point in space to another. After it arrives at its destination, the size of its displacement is:

A: either greater than or equal to B: always equal to C: either smaller than or equal to D: could be smaller or larger

than the distance it traveled.

8

Speed and velocity

•  Velocity (vector) is displacement divided by time:

•  Speed (scalar) is distance divided by time

•  In 1D we drop the vector notation but remember to keep the sign of x1 and x2.

9

Clicker question 3 Set frequency to BA

A person starts in Boulder, drives to Denver (50 km away) in 1 hour, stays in Denver 1 hour, then speeds back to Boulder in 30 minutes.

start

5inish

50kmin1hour

50kmin1/2hour

Denver

(wait1hour)

What is the average speed of the round trip?

A: 25 km/hr

B: 67 km/hr

C: 40 km/hr

D: 75 km/hr

E: none of these

Ave speed = total distance/total time = 100 km / 2.5 hours = 40 km/hr

10

Q. In 2002, Steve Fosset traveled 32000 km around the world in 13 days (1.0×106 s) in a balloon, landing at his started point. What was his average velocity?

D. 0.03 m/s C. 0.0 m/s

Set frequency to BA

A. 0.032 m/s B. 32 m/s

Since he started and ended at the same place his displacement was 0 so

Clicker question 4

11

Rank order the speeds at times 1, 2, and 3 from the slowest to the fastest

A: V1 < V2 < V3

B: V2 < V1 < V3

C: V3 < V1 < V2

D: V3 < V2 < V1

E: V2 < V3 < V1

Speed = |velocity|, the magnitude of velocity. Velocity is the slope of the "x vs t" curve. At point 3, the slope is zero (it's flat). At point 2, it has a small (negative) slope. The speed there is small and positive. At point 1, it has a large positive slope. The speed there is large. So, V3 < V2 < V1

Clicker question 5 Set frequency to BA

12

Displacement as a function of time •  Slight label change for average velocity gives

where x0 and t0 are the starting position and time. Can set t0 = 0.

•  Solving for x gives

•  If the velocity is constant: where is the initial velocity

•  If the velocity is not constant, where is the initial position at t=0

€

x = x0 + v(t)dt0

t∫

€

x0

A marathon runner runs at a steady 15 km/hr. When the runner is 7.5 km from the finish, a bird begins flying from the runner to the finish at 30 km/hr. When the bird reaches the finish line, it turns around and flies back to the runner, and then turns around again, repeating the back-and-forth trips until the runner reaches the finish line. How many kilometers does the bird travel?

13

A: 10 km B: 20 km C: 15 km E: Not sure/impossible to decide

The runner goes at 15 km/hr, and has to run 7.5 km to finish. How long does that take her? Well, 1/2 an hour (v*t = d, so t = d/v = 7.5 km / 15 km/hr = .5 hr) And during that half hour, the little bird is flying back and forth at a steady 30 km/hr. If it goes 30 km/hr * 1/2 an hour = 15 km, that's how far the bird went!

Clicker question 6 Set frequency to BA

14

x vs t graphs •  A bicyclist goes 100 m in a straight line at constant

speed in 10 s. How can we represent this motion? •  Can plot the position x versus

time t over the 10 seconds •  What is the average velocity

and speed of the bicyclist?

x(m)

t(s) 0 10

100

•  Note this is the slope of the above graph

Speed is the same

15

Non-constant velocity x(m)

t(s) 0 10

100 Constant velocity

Non-constant velocity

t(s)

v(m/s)

0 10

10

v(m/s)

t(s) 0 10

20 x(m)

t(s) 0 10

100

Slope of tangent line at each time gives instantaneous velocity

16

Clicker question 7 Q. A train moves along a straight track. The graph shows the train position as a function of time. The graph shows that the train:

D. speeds up part of the time and slows down part of the time

C. moves at constant velocity

Set frequency to BA

A. speeds up the whole time B. slows down the whole time

The slope is largest at t = 0 and continually decreases as time increases so the velocity is decreasing (but positive)

E. none of the above

17

Acceleration

so

We define acceleration as change of velocity over time

in 1D

Instantaneous acceleration is

For constant velocity,

Instantaneous velocity:

For constant velocity case: so

Acceleration is slope of velocity vs time graph

18

Position, velocity, acceleration vs t

v(m/s)

t(s) 0 10

20

x(m)

t(s) 0 10

100

t(s)

v(m/s)

0 10

10

Constant velocity

t(s)

a(m/s2)

0 10

2

x(m)

t(s) 0 10

100

Constant acceleration

t(s)

a(m/s2)

0 10

2

19

Clicker question 8 Q. An object’s velocity vs time graph is shown on the right. What best describes the car’s acceleration vs time?

Set frequency to BA

E. None of these

t

a

t

a

t

a

t

a

A B

C D

20

Clicker question 9 Set frequency to BA

A particle starts at the origin. Below is a graph of velocity vs. time.

What is the approximate position at t=3 sec?

A: 3 m B: 6 m

C: 9 m D: 18 m

E: None of these/not enough information.

x is the AREA under the "v vs t" curve. The area under the triangle is (1/2) base*height = (1/2)*3 sec *6 m/s=9 m.

Another approach is to observe that "straight line v vs t" means CONSTANT acceleration, so you can use delta x = (1/2)* a * t^2. Here, a is the slope of the line, which is 6 m/s / (3 sec) = 2 m/s^2. So Delta x = (1/2)*(2 m/s^2)*(3 sec)^2 = 9 m. Same answer, Will derive next lecture!

Summary •  Read Chapter 2 of H+R – “Motion along a

Straight Line” •  Register your Clickers •  Think about attending the talk on Saturday by Al Bartlett

21