Introduction


Discus, javelin, shot, and hammer throwing are Olympic field events involving projectiles. Other Olympic events involve jumping. For example, while competitors in the long jump frequently hurl themselves over 7 m into a sandpit, ski jumpers can travel over 100 m between take-off and landing.



The distance that a competitor can throw a projectile depends upon the strength and technique of the athlete as well as the aerodynamics of the object projected. The speed of the object before release, the angle at which it is released, and the ability to maximize the aerodynamic features of the projectile all contribute to the distance between the release and landing positions. Rotation of a javelin or discus in flight produces extra lift which prolongs flight and therefore results in longer distances.

In this consideration of projectile motion we will treat all thrown objects as small point masses and will ignore the effects of the air through which the projectile is travelling. The horizontal distance travelled by a fired or thrown object is called the range and we will show how this can be determined using the equations of motion.

The apparatus shown in Fig.1 can be used to drop the blue ball vertically while simultaneously projecting the orange ball horizontally.



The result of Fig.1 seems to defy common sense. It would seem to suggest that an archer, firing a bolt horizontally from a crossbow while at the same time dropping the crossbow, would see both the bolt and the crossbow landing on the ground at the same instant!

In practice, they would not hit the ground at the same instant because it would be difficult to synchronize releasing the bolt with dropping the bow. Additionally, the flights on the bolt are designed to give it 'lift', keeping it in the air longer. Many other projected objects adopt similar techniques to prolong flight.



In our simple model for the motion of a projectile, we have to make the following assumptions:

• Only the force of gravity acts on the vertical motion.
  
  
• No frictional forces impede the horizontal motion.
  
  

The blue ball is dropped vertically. In equal time intervals the distance it travels increases, so the ball is accelerating as it falls. The blue ball is freely falling due to the force of gravity, so its

acceleration

An object's acceleration is its rate of change of velocity.

acceleration is a constant 9.81 ms⁻².



The orange ball is projected horizontally and during its flight it travels both downwards and to the left. Its motion has both horizontal and vertical components.

Considering only the downward component of motion, we can see from the positions shown in Fig.1 that the vertical displacements of the projected ball, at any instant, are identical to those of the dropped ball. Therefore the projected ball is also accelerating downwards at 9.81 ms⁻².

The horizontal distance between successive positions is equal, so the horizontal motion is a constant speed.



The movement of the projected ball can be treated as being made up from independent horizontal and vertical motions.

• The horizontal motion is a constant speed.
  
  
• The vertical motion is a constant acceleration.
  
  

The motorcycle in Fig.3 is travelling horizontally at take-off. In flight it follows a projectile path, so we can treat its vertical and horizontal motions separately. To reach the landing-zone the bike must travel 48 m horizontally in the same time as it travels 19.62 m vertically. Use the animation to determine the horizontal speed required to land in the centre of the landing-zone.


 



Looking at its vertical motion, the motorcycle has been falling with an acceleration of 9.81 ms⁻² for a time of 2 seconds when it lands in the centre of the landing-zone.




Since the calculated horizontal and vertical distances travelled match those from the situation in Fig.3, we can see that treating the projectile's motion as separate horizontal and vertical motions accurately describes the situation.

At this point it is interesting to note that the flight time depends only on the vertical distance between take-off and landing and is independent of the take-off speed.



In this situation we have modelled the motion of the projectile as separate horizontal and vertical motions and have used the equations of motion to determine the horizontal and vertical displacements at one particular time.

Set the

velocity

An object's velocity states both the speed and direction of motion relative to a fixed reference point.

velocity of the cannon ball in Fig.4 to 13.5 ms⁻¹ and note the effect of altering the projected angle on the range.



When the cannon is fired, the ball leaves with a specific velocity. Being a

vector

A vector quantity is specified fully only when its magnitude and direction are both quoted.

vector quantity, the velocity has both size and direction.



The velocity has both horizontal and vertical components. The horizontal component of the velocity v_H is given by



The vertical component of the velocity v_V is given by



Our simple model of projectile motion states that the speed in the horizontal plane is constant, so for the entire duration of the flight the horizontal speed of the flight is where v is the projected speed and θ the angle.

Set the velocity of the cannon ball in Fig.4 to 13.5 ms⁻¹ and the angle of projection to 45°. Note the arrow in the diagram showing the components of the speed throughout the motion. While the horizontal speed of the ball in Fig.6 is constant, the speed in the vertical direction changes.



Initially, the vertical speed of the cannon ball decreases to zero and then increases again during the downward journey. As gravity is the only force acting in the vertical direction, its vertical motion is a

deceleration

The deceleration of an objects is a measurement of its rate of change of velocity as it slows down.

deceleration from its initial value to 0 ms⁻¹ at approximately 10 ms⁻² as it rises to its highest point. On the downward part of its journey, the vertical speed increases from 0 ms⁻¹ at 10 ms⁻².

In the following example, a ball is projected with a velocity of 25 ms⁻¹ at an angle of 30° to the horizontal.




By regarding the horizontal and vertical motion as separate and using the equations of motion we can determine the range of a fired projectile.

In Fig.7 the projectile's initial velocity is fixed. Alter the angle of projection to find the angle required for maximum range.



Fig.7 illustrates clearly that the range of the projectile depends on the angle of projection. Maximum range is achieved with a projection angle of 45°. This observation can be proven mathematically as follows.




Since v and g are constants for this situation, the maximum range will be achieved when the sin 2θ is maximum. The maximum value of a sine function is 1 and this happens for angles of 90°. If 2θ = 90°, then θ = 45°.

This mathematical derivation confirms the observations from Fig.7. We must remember, however, that our model for the motion of the projectile depends upon certain assumptions and is valid only while these assumptions hold.

Summary


The range of a projectile depends upon the speed before release and the angle of projection.

Projectile motion can be modelled by considering the speeds of the separate horizontal and vertical components of the motion.

The speed in the horizontal direction is constant.

The projectile's vertical motion is a uniform acceleration caused by the force of gravity.

The maximum range for any given speed is achieved for a projection angle of 45°.

The actual velocity at any instant during a projectile's flight is the vector sum of the horizontal and vertical speeds.