1. Acceleration is the change in velocity per unit time.
When an unbalanced force acts on an object it accelerates.
An unbalanced force acting on a stationary object will make it move.
An unbalanced force can make a moving object get faster or slower or change direction.
Acceleration is the change in velocity per unit time.

2. initial velocity (ms-1)
The acceleration or deceleration of an object is calculated using:
final velocity (ms-1)
acceleration
(ms-2)
initial velocity (ms-1)
time (s)

3. Measuring Acceleration
Stopwatch
Diagram
Electronic Timer
Length of Card
Light Gate
Trolley
Start Line
Finish Line

4. Method
A trolley is fitted with a length of card. The length of card is measured using a ruler.
The trolley is released from rest, so the initial velocity is 0 ms-1.
The time taken for the trolley to travel from the start to the finish line is measured using a stopwatch.
An electronic timer measures the time taken for the length of card to pass through the light beam. The timer starts when the card breaks the light beam and stops when the card leaves the light beam.
The final velocity is calculated using:
These measurements are used to calculate acceleration in the relationship shown:

5. Results

6. Motion Graphs There are three types of motion graph we will study:
A velocity - time graph.
A displacement – time graph.
An acceleration – time graph.

7. displacement-time
velocity-time
acceleration-time
Constant Velocity t s t v t a
Constant Acceleration t s t v t a

8. Constant Decelerationv t a

9. Drawing Graphs
Each velocity – time graph has a corresponding acceleration – time graph and displacement- time graph.
Example
The following velocity – time graph describes a journey.
time / s
velocity / ms-1 3 10 12

10. Draw the corresponding acceleration – time graph.
0-3 seconds
3-10 seconds
10-12 seconds

11. time / s
acceleration / ms-2 3 4 10 12 -6

12. Q1. Copy out the following velocity - time graph and underneath it
Q1. Copy out the following velocity - time graph and underneath it draw the corresponding acceleration – time graph.
time / s
velocity / ms-1 4 13 16 20 9 24

13. 0-4 seconds
4-9 seconds
9-13 seconds
13-16 seconds

14. acceleration / ms-2
time / s 4 5 9 13
- 8 16 1

15. Q2. Using the following acceleration – time graph of an object
Q2. Using the following acceleration – time graph of an object starting from rest, draw the corresponding velocity – time graph.
acceleration / ms-2 5 -2 4 8 18
time / s

16. velocity / ms-1 20 4 8 18
time / s

17. Q3. Copy out the following velocity – time graph and underneath it
Q3. Copy out the following velocity – time graph and underneath it draw the corresponding acceleration – time graph (after appropriate calculations).
velocity / ms-1
time / s 3 15 -9 7 9 14 24 18

18. 0-3 seconds
3-7 seconds
7-9 seconds
9-14 seconds
14-18 seconds
18-24 seconds

19. acceleration / ms-2 4 -6 3 7 9 14 24 18
time / s

20. Motion and Direction
In the velocity-time graphs so far, motion has been in only one direction.
A velocity time graph however can show two different directions; one direction is positive, while the other is negative.

21. Example 1
A car travelling along a straight section of road has the following velocity-time graph.
forward motion
backward motion
(a) Calculate the distance travelled by the car.
(b) Calculate the displacement of the car.

22. (a) Distance travelled is the area under velocity-time graph.
(b) Displacement is the forwards motion, less the backward motion.

23. Worksheet – Graphs of Motion
Q1 – Q9

24. Ball Thrown Into Air A ball is thrown directly upwards into the air.
It rises into the air and falls back down to the thrower.
The velocity – time graph and corresponding acceleration – time graph are shown.
velocity / ms-1
time / s

25. acceleration / ms-2
time / s
-9.8

26. Bouncing Ball (No Energy Loss)
A ball is dropped from a height to the ground.
The ball bounces twice with no energy loss and is then caught.
The velocity - time and acceleration - time graphs are as follows:
max height
time / s
velocity / ms-1
rising
rising
falling
falling
downwards (falling): - ve
hits ground
upwards (rising): + ve

27. time / s
acceleration / ms-2
-9.8

28. falling falling rising rising max height
TASK
Re-draw the velocity-time graph for the bouncing ball, taking downward motion as positive, and upward motion as negative.
hits ground
time / s
velocity / ms-1
downwards (falling): + ve
upwards (rising): - ve
falling
falling
rising
rising
max height

29. Bouncing Ball (With Energy Loss)
A ball is dropped from a height to the ground.
Kinetic energy is lost with each bounce.
The velocity - time and acceleration - time graphs are as follows:
time / s
velocity / ms-1

30. time / s
acceleration / ms-2
-9.8
TASK
Re-draw the velocity-time graph for the bouncing ball, taking downward motion as positive, and upward motion as negative.

31. Worksheet – More Graphs of Motion
Q1 – Q7

32. Equations of Motion There are three equations of motion.
You must be able to use and derive these three equations of motion.
1st Equation of Motion

33. Displacement (s) is the area under a velocity time graph.
2nd Equation of Motion
To derive the second equation of motion, the velocity-time graph shown is used as a starting point.
time/s
velocity/ms-1 v
v - u u t
Displacement (s) is the area under a velocity time graph.

34. but from equation (1):
so we can rewrite as:

35. This equation links final velocity (v) and displacement (s).
3rd Equation of Motion
This equation links final velocity (v) and displacement (s).
taking a common factor of 2a gives
and since s = ut + ½at2

36. Direction
When using the equations of motion, it is essential that direction is considered.
In these examples, upward motion is taken as positive, so any downward motion is taken to be negative.
Example
A helicopter is travelling upwards with a velocity of 25 ms-1.
A package is released and hits the ground 14 s later.
Path of Package
stationary ( 0 ms-1 ) *
released
hits ground

37. Choose the upward direction as positive!
This example has motion in two directions.
It is necessary to distinguish between the two directions.
Choose the upward direction as positive!
(a) How long will it take the package to reach its maximum height? (2)

38. The negative indicates travelling downwards.
(b) How high as it climbed since being released? (2)
(c) Calculate the velocity of the package just before it hits the ground (2)
The negative indicates travelling downwards.

39. (d). How high above the ground is the helicopter when the
(d) How high above the ground is the helicopter when the package is released? (2)
So the helicopter is m above the ground.

40. Worksheet – Equations of Motion
Q1 – Q15

41. Acceleration Due To Gravity
Diagram
Timer
Ball Bearing
Trap Door

42. Method
The ball bearing is released from rest, so initial velocity (u) is 0 ms-1.
The displacement (s) of the ball bearing is the distance between the release point and trap door, and is measured using a metre stick.
An electronic timer measures the length of time taken for the ball bearing to reach the trap door.
Calculation

43. Projectiles (Half Trajectory)
An object projected sideways through the air will follow a curved trajectory.
horizontal motion (steady speed)
vertical motion
accelerates downwards at -9.8 ms-2
The horizontal and vertical motions should be treated separately.
Time is the only quantity common to both.

44. vh
This is an example of a
‘half-trajectory.’ vv
GREEN – actual motion
RED – vertical motion
BLUE – horizontal motion

45. resultant/actual velocity
At any point in its trajectory, the velocity of a projectile has two components.
one vertical, VV
the other horizontal, VH
The resultant velocity is found drawing a vector diagram and add the vectors together, TIP to TAIL.
Vector Diagram
horizontal velocity
vertical velocity
resultant/actual velocity

46. common to horizontal and vertical motions
Example
A ball is kicked horizontally off an embankment, with a velocity of 30 ms-1.
It lands 24 m from the base of the embankment.
(a) Calculate how long the ball was in flight.
30 ms-1
24 m
common to horizontal and vertical motions

47. (b) Calculate the horizontal velocity just before hitting the ground.
acted upon by gravity
Horizontal
Vertical
not initially falling down, so speed of zero in vertical direction
travels horizontally at steady speed – no acceleration horizontally
Horizontal

48. means ball fell through distance of 3.14 m
(c) Calculate the vertical velocity just before hitting the ground.
Vertical
means 7.84 ms-1 downwards
(d) How high is the embankment?
Vertical
means ball fell through distance of 3.14 m
so height of the embankment is 3.14 m

49. (e). Calculate the resultant velocity of the ball, just before hitting
(e) Calculate the resultant velocity of the ball, just before hitting the ground.
30 ms-1
-7.8 ms-1
velocity
Size
By Pythagoras:

50. Direction
30 ms-1
-7.8 ms-1
velocity

51. You may want to draw a diagram to help you get started !!!
Q1. A ball is kicked off a cliff with a horizontal speed of 16 ms-1.
The ball hits the ground 2.2 s later.
(a) Calculate the height of the cliff.
(b) Calculate the distance between the foot of the cliff and where the ball lands.
(c) Calculate the vertical component of the balls velocity just before it hits the ground.
(d) Calculate the balls velocity as it hits the ground.
23.7 m
35.2 m
21.6 ms-1
26.9 ms-1 at angle of 53.5° below horizon
You may want to draw a diagram to help you get started !!!

52. You may want to draw a diagram to help you get started !!!
Q2. A ball is kicked off a cliff with a horizontal speed of 22 ms-1. the ball hits the ground 1.5 s later.
(a) Calculate the height of the cliff.
(b) Calculate the horizontal distance from the foot of the cliff, to where the ball lands.
(c) Calculate the vertical component of the balls velocity as it hits the ground.
(d) Calculate the balls actual velocity as it hits the ground.
11 m
33 m
14.7 ms-1
26.5 ms-1 at angle of 34° below horizon
You may want to draw a diagram to help you get started !!!

53. Does Projectile Theory Work?
Diagram
ball-bearing
sand h d

54. Measurements
Horizontal Velocity measure distance ball-bearing travels along desk and divide by time taken
Vertical Displacement measure height of desk from floor
Calculation
Calculate the time of flight.
Vertical

55. Now calculate the horizontal displacement.
Experimentally
The horizontal displacement was measured experimentally using a metre stick to be m.

56. Projectiles (Full Trajectory)
A projectile does not need to be an object falling, but could be an object fired at angle to the horizontal. θ
The subsequent motion would be
max height

57. Horizontal a = 0 ms-2 Vertical a = -9.8 ms-2
If air resistance is ignored, the trajectory has an axis of symmetry about the mid point (maximum height).
So the time taken to reach the maximum height is the same as the time taken to fall back to the ground.
Various calculations can be made, but firstly, the initial velocity must be split into its horizontal and vertical components.
Horizontal
a = 0 ms-2
Vertical
a = -9.8 ms-2

58. Example 1
A golf ball is hit off the tee at 48 ms-1 at angle of 20° to the horizontal.
20°
48 ms-1
Calculate the horizontal and vertical components of the initial velocity.
Horizontal
Vertical
48 ms-1
20° VH VV

59. Example 2
An arrow is projected into the air with a velocity of 38 ms-1 at an angle of 25° to the horizontal.
38 ms-1
250
(a) Calculate the horizontal and vertical components of the initial velocity.
38 ms-1
25° VH VV
Horizontal
Vertical

60. (b) Calculate the arrow’s maximum height.
Vertical

61. (c). Calculate the time taken for the arrow to reach its maximum
(c) Calculate the time taken for the arrow to reach its maximum height.
Vertical
(d) Calculate the total time of the arrows flight.

62. (e). Calculate the horizontal distance travelled by the arrow until
(e) Calculate the horizontal distance travelled by the arrow until impact with the ground.
Horizontal

63. (f) Calculate the arrow’s velocity 0.5 s after being fired.
Firstly calculate the vertical component of velocity (horizontal component is constant, since a = 0 ms-2)
Vertical
Now calculate the actual velocity after combining the vertical and horizontal components of the velocity after 0.5 s.

64. Velocity of the arrow after 0.5 s is:
Size
By Pythagoras: v θ
34.4 ms-1
11.2 ms-1
Direction
Velocity of the arrow after 0.5 s is:
36.2 ms-1 at angle of 18° above the horizon

65. 53.7 ms-1 at angle of 48° above the horizon
Q1. A shell is fired from a gun with a velocity of 72 ms-1 at an angle of 60° to the horizontal.
(a) Calculate the horizontal and vertical components of the initial velocity.
(b) Calculate the maximum height reached.
(c) Calculate the time taken for the shell to reach it’s maximum height.
(d) Calculate the total time of flight.
(e) Calculate the horizontal range of the shell.
(f) Calculate the shells velocity after 2.3 s.
VH = 36 ms-1
VV = 62.4 ms-1
199 m
6.4 s
12.8 s
458 m
53.7 ms-1 at angle of 48° above the horizon

66. 45.89 ms-1 at angle of 19.3° above horizon
Q2. An arrow is fired with a velocity of 50 ms-1 at an angle of 30° to the ground.
(a) Calculate the time taken for the arrow to reach its maximum height.
(b) Calculate the maximum height reached by the arrow.
(c) Calculate the time the arrow is in flight.
(d) Calculate how far away from the firing point the arrow will land.
(e) Calculate the actual velocity of the arrow 1s after it is fired.
2.55 s
31.89 m
5.1 s
5.1 s
45.89 ms-1 at angle of 19.3° above horizon