1. Lesson 2.2 Acceleration
Essential Question: How do you calculate the acceleration of an object?

2. Describe the speedometer on a car that is entering a highway, drives for 25 miles and then exits the highway.

3. Consider the two situations below
When a shuttle bus approaches a stop, the driver begins to apply the brakes to slow down 5.0 s before actually reaching the stop. The speed changes from 9.0 m/s to 0 m/s over time.
A shuttle bus driver has to slam on his brakes to avoid hitting a dog. The bus slows from 9.0 m/s to 0 m/s in just 1.5 s.

4. What is acceleration? A measure of how quickly your velocity changes
The rate at which velocity changes over time

5. An object accelerates if its speed, direction or both change.
Acceleration has dimensions of length divided by time squared. It’s SI units are meters per second squared.

6. Acceleration Uniform Nonuniform Unchanging acceleration
The acceleration due to gravity near the surface of the Earth
Change in acceleration
When you’re driving, you encounter stop signs or stop lights often, you slow to a stop and then speed up again

7. How do you calculate the acceleration of an object?
𝑎 𝑎𝑣𝑔 = ∆𝑣 ∆𝑡 = 𝑣 𝑓 − 𝑣 𝑖 𝑡 𝑓 − 𝑡 𝑖 average acceleration= change in velocity time required for change

8. A shuttle bus slows down with an average acceleration of −1. 8 m s 2
A shuttle bus slows down with an average acceleration of −1.8 m s 2 . How long does it take the bus to slow from 9.0 m/s to a complete stop?

9. As the shuttle bus comes to a sudden stop to avoid hitting a dog, it accelerates uniformly at −4.1 m s 2 as it slows from 9.0 m/s to 0.0 m/s. Find the time interval of acceleration for the bus.

10. A car traveling at 7. 0 m/s accelerates uniformly at 2
A car traveling at 7.0 m/s accelerates uniformly at 2.5 m s 2 to reach a speed of 12.0 m/s. How long does it take for this acceleration to occur?

11. With an average acceleration of −1
With an average acceleration of −1.2 m s 2 , how long will it take a cyclist to bring a bicycle with a initial speed of 6.5 m/s to a complete stop?

12. Turner’s treadmill runs with a velocity of −1
Turner’s treadmill runs with a velocity of −1.2 m/s and speeds up at regular intervals during a half-hour workout. After 25 min, the treadmill has a velocity of −6.5 m/s. What is the average acceleration of the treadmill during this period?

13. Supposed a treadmill has an average acceleration of 4.7× 10 −3 m s 2 .
How much does its speed change after 5.0 min?
If the treadmill’s initial speed is 1.7 m/s, what will its final speed be?

14. Is acceleration just a number?
No.
Acceleration has direction and magnitude.

15. Is acceleration positive or negative?
Acceleration can be positive or negative.
The sign of acceleration tells you whether you’re speeding up or slowing down.
The sign of acceleration depends on direction.

16. A negative value for acceleration does not always indicate a decrease in speed.
An object could be moving in a negative direction, so a negative acceleration would indicate an increase in speed.

17. If the final velocity is…
Greater than the initial velocity, ∆𝑣 will be positive and the acceleration will be positive.
The same as the initial velocity, ∆𝑣 is 0 m/s. When velocity is constant, the acceleration is equal to 0.
Less than the initial velocity, ∆𝑣 will be negative and the acceleration will be negative.

18. Velocity v. Time Graph
The slope and shape of the velocity-time graph can describe a object’s motion.

19. When the velocity in the positive direction is increasing, the acceleration is positive, as at point A. When the velocity is constant, there is no acceleration, as at point B. When the velocity in the positive direction is decreasing, the acceleration is negative, as at point C.

20. Velocity and Acceleration
𝒗 𝒊 𝒂
Motion +
Speeding up -
Slowing down
- or +
Constant velocity
Speeding up from rest
Remaining at rest

21. Motion with Constant Acceleration
The time interval between successive images is 0.10 s.
The displacement between each image is greater.
The velocity increases by exactly the same amount during each time interval.
The acceleration is constant.

23. How does displacement relate?
The displacement of an object is dependent upon acceleration, initial velocity and time.

24. Displacement with Constant Acceleration
∆𝑥= 1 2 𝑣 𝑖 + 𝑣 𝑓 ∆𝑡 displacement= 1 2 initial velocity+final velocity initial velocity+final velocity time interval

25. A racing car reaches a speed of 42 m/s
A racing car reaches a speed of 42 m/s. It then begins a uniform negative acceleration, using its parachute and braking system, and comes to rest 5.5 s later. Find the distance that the car travels during braking.

26. A bicyclist accelerates from 5. 0 m/s to 16 m/s in 8. 0 s
A bicyclist accelerates from 5.0 m/s to 16 m/s in 8.0 s. Assuming uniform acceleration, what distance does the bicyclist travel during this time interval?

27. A car accelerates uniformly from rest to a speed of 6. 6 m/s in 6. 5 s
A car accelerates uniformly from rest to a speed of 6.6 m/s in 6.5 s. Find the distance the car travels during this time.

28. When Maggie applies the brakes of her car, the car slows uniformly from 15.0 m/s to 0.0 m/s in 2.50 s. How many meters before a stop sign must she apply her brakes in order to stop at the sign?

29. A driver in a car traveling at a speed of 21
A driver in a car traveling at a speed of 21.8 m/s sees a cat 101 m away on the road. How long will it take for the car to accelerate uniformly to a stop in exactly 99 m?

30. A car enters the freeway with a speed of 6
A car enters the freeway with a speed of 6.4 m/s and accelerates uniformly for 3.2 km in 3.5 min. How fast (in m/s) is the car moving after this time?

31. You are being strapped into a jet on an aircraft carrier deck
You are being strapped into a jet on an aircraft carrier deck. The mechanic says you need to take off at a speed of at least 62.0 m/s. You’ll be catapulted at an acceleration of 31 m s 2 . Is there going to be enough catapult to do the job? The catapult is 100 meters long.

32. Can you calculate displacement if you don’t know the final velocity?
Yes you can. You can manipulate the acceleration formula to find an expression for final velocity which can be used in the other equation.

33. Velocity with Constant Acceleration
𝑣 𝑓 = 𝑣 𝑖 +𝑎∆𝑡 final velocity=initial velocity+ acceleration×time interval

34. Displacement with Constant Acceleration
∆𝑥= 𝑣 𝑖 ∆𝑡+ 1 2 𝑎 ∆𝑡 2 displacement= initial velocity×time interval acceleration× time interval 2

35. A plane starting at rest at one end of a runway undergoes a uniform acceleration of 4.8 m s 2 for 15 s before takeoff. What is its speed at takeoff? How long must the runway be for the plane to be able to take off?

36. A car with an initial speed of 6
A car with an initial speed of 6.5 m/s accelerates at a uniform rate of m s 2 for 3.6 s. Find the final speed and the displacement of the car during this time.

37. An automobile with an initial speed of 4
An automobile with an initial speed of 4.30 m/s accelerates uniformly at the rate of m s 2 . Find the final speed and the displacement after 5.00 s.

38. A car starts from rest and travels for 5
A car starts from rest and travels for 5.0 s with a constant acceleration of −1.5 m s 2 . What is the final velocity of the car? How far does the car travel in this time interval?

39. A driver of a car traveling at 15
A driver of a car traveling at 15.0 m/s applies the brakes, causing a uniform acceleration of −2.0 m s 2 . How long does it take the car to accelerate to a final speed of 10.0 m/s? How far has the car moved during the braking period?

40. You become a drag racer. After a test race, you know the distance you went, 402 meters, and you know the time it took, 5.5 seconds. What was your acceleration as you blasted down the track?

41. You become a drag racer. Your acceleration is 26
You become a drag racer. Your acceleration is m s 2 and your final speed is m/s. Find the total distance you traveled.

42. There you are, the Tour de France hero, ready to give a demonstration of your bicycling skills. There will be a time trial of 8.0 seconds. Your initial speed is 6.0 m/s, and when the whistle blows you accelerate at 2.0 m s 2 for the 8.0 seconds allowed. At the end of the time trial, how far will you have traveled?

43. What if there is no time?
It is possible to find a final velocity, an initial velocity or an acceleration without the time interval.
The final velocity depends on the initial velocity, acceleration and the displacement.

44. Final Velocity after any Displacement
𝑣 𝑓 2 = 𝑣 𝑖 2 +2𝑎∆𝑥 final velocity 2 = initial velocity 2 +2 acceleration displacement

45. A person pushing a stroller starts from rest, uniform accelerating at a rate of m s 2 . What is the velocity of the stroller after it has traveled 4.75 m?

46. An aircraft has a landing speed of 83. 9 m/s
An aircraft has a landing speed of 83.9 m/s. The landing area of an aircraft carrier is 195 m long. What is the minimum uniform acceleration required for a safe landing?

47. A person pushing a stroller starts from rest, uniform accelerating at a rate of m s 2 . What is the velocity of the stroller after it has traveled 6.32 m?

48. A car traveling initially at +7
A car traveling initially at +7.0 m/s accelerates uniformly at the rate of m s 2 for a distance of 245 m.
What is its velocity at the end of the acceleration?
What is its velocity after it accelerates for 125 m?
What is its velocity after it accelerates for 67 m?

49. A car accelerates uniformly in a straight line from rest at the rate of 2.3 m s 2 .
What is the speed of the car after it has traveled 55 m?
How long does it take the car to travel 55 m?

50. A motorboat accelerates uniformly from a velocity of 6
A motorboat accelerates uniformly from a velocity of 6.5 m/s to the west to a velocity of 1.5 m/s to the west. If its acceleration was 2.7 m s 2 to the east, how far did it travel during the acceleration?

51. An aircraft has a liftoff speed of 33 m/s
An aircraft has a liftoff speed of 33 m/s. What minimum constant acceleration does this require if the aircraft is to be airborne after a take-off run of 240 m?

52. A certain car is capable of accelerating at a uniform rate of 0
A certain car is capable of accelerating at a uniform rate of m s 2 . What is the magnitude of the car’s displacement as it accelerates uniformly from a speed of 83 km/h to one of 94 km/h?

53. Equations for Constantly Accelerated Straight-Line Motion
Form to use when accelerating object has an initial velocity
Form to use when accelerating object starts from rest
∆𝑥= 𝑣 𝑖 + 𝑣 𝑓 ∆𝑡
∆𝑥= 1 2 𝑣 𝑓 ∆𝑡
𝑣 𝑓 = 𝑣 𝑖 +𝑎∆𝑡
𝑣 𝑓 =𝑎∆𝑡
∆𝑥= 𝑣 𝑖 ∆𝑡+ 1 2 𝑎 ∆𝑡 2
∆𝑥= 1 2 𝑎 ∆𝑡 2
𝑣 𝑓 2 = 𝑣 𝑖 2 +2𝑎∆𝑥
𝑣 𝑓 2 =2𝑎∆𝑥

54. How do you calculate the acceleration of an object?