Acceleration 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.

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

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

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:

Results

Motion Graphs There are three types of motion graph we will study: 1.A velocity - time graph. 2.A displacement – time graph. 3.An acceleration – time graph.

displacement-timevelocity-timeacceleration-time Constant Velocity Constant Acceleration t s t v t a t s t v t a

Constant Deceleration t a t v t s

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

Draw the corresponding acceleration – time graph. 0-3 seconds3-10 seconds10-12 seconds

time / s acceleration / ms

Q1.Copy out the following velocity - time graph and underneath it draw the corresponding acceleration – time graph. time / s velocity / ms

0-4 seconds4-9 seconds9-13 seconds seconds

acceleration / ms -2 time / s

Q2.Using the following acceleration – time graph of an object starting from rest, draw the corresponding velocity – time graph. acceleration / ms time / s

velocity / ms time / s

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

0-3 seconds3-7 seconds7-9 seconds 9-14 seconds14-18 seconds seconds

acceleration / ms time / s

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.

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

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

Worksheet – Graphs of Motion Q1 – Q9

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

acceleration / ms -2 time / s -9.8

falling 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: time / s velocity / ms -1 downwards (falling): - ve upwards (rising): + ve rising falling rising hits ground max height

time / s acceleration / ms

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

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

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

Worksheet – More Graphs of Motion Q1 – Q7

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

2 nd 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 u v t v - u Displacement (s) is the area under a velocity time graph.

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

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

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. * * * released stationary ( 0 ms -1 ) hits ground Path of Package

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)

(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.

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

Worksheet – Equations of Motion Q1 – Q15

Acceleration Due To Gravity Diagram Ball Bearing Trap Door Timer

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

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

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

At any point in its trajectory, the velocity of a projectile has two components. one vertical, V V the other horizontal, V H 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

30 ms -1 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. 24 m common to horizontal and vertical motions

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

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

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

Direction 30 ms ms -1 velocity

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 m 35.2 m 21.6 ms ms -1 at angle of 53.5° below horizon 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 14.7 ms ms -1 at angle of 34° below horizon 33 m You may want to draw a diagram to help you get started !!!

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

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

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

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

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

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

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 (a)Calculate the horizontal and vertical components of the initial velocity. Horizontal Vertical 38 ms -1 25° VHVH V

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

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

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

(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.

v θ 34.4 ms ms -1 Size By Pythagoras: Direction Velocity of the arrow after 0.5 s is: 36.2 ms -1 at angle of 18° 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. V H = 36 ms -1 V V = 62.4 ms m 6.4 s 12.8 s 458 m 53.7 ms -1 at angle of 48° above the 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 s m 5.1 s ms -1 at angle of 19.3° above horizon