Topic 9.2

1.2.1 – Projectile Motion

Free Fall Motion

Any object moving freely under gravity will accelerate downwards.

Consider the energy change in this situation. GPE

lost = KE

gained

mgΔh = ½ mv2

2gh = v2

As this equation has no mass term ALL OBJECTS OF THE SAME SIZE WILL FALL AT THE SAME RATE REGARDLESS OF MASS!

What is a Projectile

A projectile is any object that is launched and then moves ONLY under the influence of gravity.

A ball becomes a projectile once struck. A bullet, bomb or shell is a projectile. A rocket or missile is NOT a projectile whilst

still under power.

Projectile Motion

Consider two identical objects. One is dropped and one is projected horizontally. Which one hits the ground first?

Projectile Motion

This demonstration shows that THE HORIZONTAL AND VERTICAL COMPONENTS OF MOTION ARE INDEPENDENT.

This means that the standard uvast equations learnt in the preliminary course can be used separately in both the horizontal and vertical directions.

Galileo

• It was Galileo who first discovered that two objects falling freely accelerated at the same rate.

• He (allegedly) dropped a cannon ball and cabbage from the leaning tower of Pisa.• The two objects hit the ground together!

• In order to slow the motion down further, he rolled cannonballs down a smooth incline.• He saw that their motion was made up of two parts.

• Horizontal at constant velocity• Vertical at constant acceleration

The uvast equations

For a body undergoing constant acceleration...

2

2

22

2

12

12

2as

atvt=s

at+ut=s

tu+v

=a

+u=v

at+u=v

Projectile Motion

Vertically Upwards is +ve Acceleration is

9.8ms-2

downwards a = g = -9.8ms-2

Horizontally

Rightwards is +ve

Acceleration is 0ms-2

Assuming there is no air resistance!

Projectile Motion

As there is only acceleration in one direction the flight path (trajectory) of a projectile is a parabola.

Solving Projectile Problems

Use the following method to solve projectile problems.

1) Draw a large, clear diagram.

2) Divide your page into two; vertical and horizontal.

3) Write down all values for u,v,a,s and t in each direction.

4) Solve the problem in the horizontal and vertical directions remembering that t is the same for both!

Example 1

A cannon launches a shell horizontally at 50ms-1 from the top of a 25m high castle wall. What is the time of flight? What is the horizontal range of the

ball?

Example 2

A cliff diver projects himself with a horizontal velocity of 4ms-1 from the top of a 75m tall cliff. How long does it take him to hit the

water? How far from the base of the cliff does

he land? What is his velocity when he enters

the water?

Full Projectile Motion

For a projectile not launched horizontally, the vertical and horizontal components must be calculated before being used in the equations.

By trigonometry: u

y = u Sin θ

ux = u Cos θ

u

ux

uy

θ

Full Projectile Motion

Calculate the horizontal and vertical components of the velocity of: A cannon ball fired at 300ms-1 at 25o

above the horizontal. An arrow shot at 110ms-1 at 75o

above the horizontal. A javelin thrown at 45ms-1 at 40o

above the horizontal.

Full Projectile Motion

In solving these problems, use exactly the same method as before. Also remember that:

You must resolve the velocity into 2 components.

At the highest point vy = 0ms-1

The motion is symmetrical so the time of flight is twice the time to the highest point.

Example 1

A golf ball is struck on a horizontal course at a velocity of 65ms-1 at 30o above the horizontal.

Calculate The time of flight. The range of the ball The maximum height of the ball.

Example 2

A catapult launches a stone at 80ms-1 at an angle of 50o above the horizontal on a level field.

Calculate The time of flight. The range of the ball The maximum height of the ball.

Example 3

A catapult launches a stone at 80ms-1 at an angle of 50o above the horizontal from the top of a 10m high hill.

Calculate The time of flight. The range of the ball The maximum height of the ball.