1. Measurement and Motion
Acceleration in a Straight Line
Acceleration
A Model for Accelerated Motion
Free Fall and the Acceleration due to Gravity

2. Objectives
Calculate acceleration from the change is speed and the change in time.
Give an example of motion with constant acceleration.
Determine acceleration from the slope of the speed versus time graph.
Calculate time, distance, acceleration or speed when given three of the four variables.
Solve two-step accelerated motion problems.
Calculate height, speed, or time of flight in free fall problems.
Explain how air resistance makes objects of different masses fall with different accelerations.

3. Vocabulary Terms initial speed acceleration free fall m/sec2
acceleration due to gravity (g)
time of flight
friction
air resistance
terminal speed
acceleration
m/sec2
delta D
constant acceleration
uniform acceleration
slope
term

4. Acceleration
Key Question:
How is the speed of the ball changing?

5. Acceleration of a car
Acceleration is the rate of change in the speed of an object.

6. Acceleration vs. Speed
Positive acceleration and positive speed

7. Acceleration vs. Speed
Negative acceleration and positive speed

8. Change in speed (m/sec)
Acceleration
Change in speed (m/sec)
Acceleration
(m/sec2)
a = Dv Dt
Change in time (sec)

9. Calculate Acceleration
A student conducts an
acceleration experiment by coasting a bicycle down a steep hill.
The student records the speed of the bicycle every second for five seconds.
Calculate the acceleration of the bicycle.
1) You are asked for the acceleration.
2) You are given a table of speeds and times from an experiment.
3) a = (v2-v1) ÷ (t2 - t1)
4) Choose any two pairs of speed and time data.
a = (6 m/sec - 4 m/sec) ÷ (3 sec - 4 sec) = 2 m/sec2
For this experiment, the acceleration would have been the same for any two points.

10. Acceleration and Speed
Constant acceleration is different from constant speed.
Motion with zero acceleration appears as a straight horizontal line on a speed versus time graph.
zero acceleration
constant speed

11. Acceleration and Speed
Constant acceleration is sometimes called uniform acceleration.
A ball rolling down a straight ramp has constant acceleration.
constant acceleration
increasing speed

12. Acceleration and Speed
An object can have acceleration, but no speed.
Consider a ball rolling up a ramp.
As the ball slows down, eventually its speed becomes zero.
constant negative
acceleration
decreasing speed

13. Slope and Acceleration
Use slope to recognize when there is acceleration in speed vs. time graphs.
Level sections (A) on the graph show an acceleration of zero.
The highest acceleration (B) is the steepest slope on the graph.
Sections that slope down (C) show negative acceleration (slowing down).

14. A Model for Accelerated Motion
Key Question:
How do we describe and predict accelerated motion?
*Students read Section 4.2 AFTER Investigation 4.2

15. Slope of a graph
The slope of a graph is equal to the ratio of rise to run.
On the speed versus time graph, the rise and run have special meanings, as they did for the distance versus time graph.
The rise is the amount the speed changes.
The run is the amount the time changes.

16. Acceleration and slope
Acceleration is the change in speed over the change in time.
The slope of the speed versus time graph is the acceleration.

17. Calculate acceleration
The following graph shows the speed of a bicyclist going over a hill.
Calculate the maximum acceleration of the cyclist and say when in the trip it occurred.
1) You are asked for the acceleration.
2) You are given a graph of speed versus time.
3) a = slope of graph
4) The steepest slope is between 60 and 70 seconds, when the speed goes from 2 to 9 m/sec.
a = (9 m/sec - 2 m/sec) ÷ (10 sec) = 0.7 m/sec2

19. Solving Motion Problems

20. Solving Motion Problems

21. Calculate speed A ball rolls at 2 m/sec onto a ramp.
The angle of the ramp creates an acceleration of 0.75 m/sec2.
Calculate the speed of the ball 10 seconds after it reaches the ramp.
1) You are asked for the speed.
2) You are given an initial speed acceleration, and time.
3) v = vo + at
4) v = 2 m/sec + (.75 m/sec2)(10 sec) = 9.5 m/sec

22. Solving Motion Problems
initial position
distance if at constant speed
distance to add or subtract, depending on acceleration

23. Calculate position
A ball traveling at 2 m/sec rolls onto a ramp that tilts upward.
The angle of the ramp creates an acceleration of -0.5 m/sec2.
How far up the ramp does the ball get at its highest point?
(HINT: The ball keeps rolling upward until its speed is zero.)
1) You are asked for distance.
2) You are given an initial speed and acceleration. You may assume an initial position of 0.
3) v = vo + at
x = xo + vot + 1/2at2
4) At the highest point the speed of the ball must be zero.
0 = 2 m/sec - 0.5t
t = 4 seconds
Now use the time to calculate how far the ball went.
x = (2 m/sec)(4 sec) - (0.5) (.5 m/sec2) (4 sec)2 = 4 meters
At its highest point, the ball has moved 4 meters up the ramp.

24. Solving Motion Problems

25. Calculate time A car at rest accelerates at 6 m/sec2.
1) You are asked for time and speed.
2) You are given vo = 0,
x = 440m, and a = 6 m/sec2 assume xo = 0
3) v = vo + at x = xo + vot + 1/2at2
4) Since xo = vo = 0, the position equation reduces to: x = 1/2at2
440 m = - (0.5)(6 m/sec2) t2
t2 = 440 ÷ 3 = 146.7
t = 12.1 seconds
Now use the time to calculate the speed.
v = (6 m/sec2)(12.1 sec) = 72.6 m/sec
This is 162 miles per hour.
A car at rest accelerates at 6 m/sec2.
How long does it take to travel 440 meters (about a quarter-mile) and how fast is the car going at the end?

26. Calculate position
A ball starts to roll down a ramp with zero initial speed.
After one second, the speed of the ball is 2 m/sec.
How long does the ramp need to be so that the ball can roll for 3 seconds before reaching the end?
1) You are asked to find position (length of the ramp).
2) You are given vo = 0, v = 2m/sec at t = 1 sec, t = 3 sec at the bottom of the ramp,
and you may assume xo = 0.
3) After canceling terms with zeros, v = at and x = 1/2at2
4) This is a two-step problem.
First, you need the acceleration, then you can use the position formula to find the length of the ramp.
a = v ÷ t = (2 m/sec) ÷ (1 sec) = 2 m/sec2
x = 1/2at2 = (0.5)(2 m/sec2)(3 sec)2 = 9 meters

28. Solving Problems with Free Fall

29. Calculate height
A stone is dropped down a well and it takes 1.6 seconds to reach the bottom.
How deep is the well?
You may assume the initial speed of the stone is zero.
1) You are asked for distance.
2) You are given an initial speed and time of flight.
3) v = vo - gt y = yo + vot - 1/2gt2
4) Since yo = vo = 0, the height equation reduces to: y = - 1/2gt2
y = - (0.5)(9.8 m/sec2)
× (1.6 sec)2
y = meters
The negative sign indicates the height is lower than the initial height by 12.5 m.

30. Air Resistance and Mass
The acceleration due to gravity does not depend on the mass of the object which is falling.
Air creates friction that resists the motion of objects moving through it.
All of the formulas and examples discussed in this section are exact only in a vacuum (no air).

31. Terminal Speed
You may safely assume that a = g = 9.8 m/sec2 for speeds up to several meters per second.
The resistance from air friction increases as a falling object’s speed increases.
Eventually, the rate of acceleration is reduced to zero and the object falls with constant speed.
The maximum speed at which an object falls when limited by air friction is called the terminal speed.

32. Free Fall and Acceleration due to Gravity
Key Question:
How do you measure the acceleration of a falling object?
*Students read Section 4.3 BEFORE Investigation 4.3

33. Application: Antilock Brakes