An object that has an initial horizontal velocity and falls freely under gravity is called a projectile. The object could be falling, launched, thrown, fired or pitched. Examples of a projectile include a ball kicked at an angle, a rock thrown off a cliff or a cannon ball fired into the air.
Trajectory – this is the name given to the path followed by a projectile.
Graph 1 shows an example of the trajectory of a projectile:
It’s important to note that the starting point of a projectile isn’t always on the ground, it could be higher up such as the top of a cliff.
Projectiles have both horizontal and vertical motion and they are independent of each other. This means that we can make separate calculations for the horizontal and vertical motion:
Horizontal Motion
The horizontal motion of a projectile has constant velocity. This means that we don’t need to consider acceleration and the only variables we need to look at are displacement, velocity and time. As these are the only three variables we need to consider the main equation used in horizontal motion calculations is:
If plotted on a graph the horizontal motion is along the x axis, as shown by graph 1. It is therefore useful to denote the quantities related to the horizontal motion with an x subscript, for example Sx = horizontal displacement.
Vertical Motion
Unlike horizontal motion the vertical motion does not have constant velocity. The vertical motion experiences constant acceleration due to gravity (g). Due to the motion having constant acceleration we can use SUVAT equations to find out useful information about the vertical motion.
As with the horizontal motion it is useful to denote the vertical quantities with a subscript. This makes it clear which motion is which. On graph 1 you can see that the vertical motion can be plotted along the y axis so we can use a y subscript, for example Sy = vertical displacement.
Horizontally or at an Angle
Projectiles can be launched horizontally or at an angle to the horizontal.
Horizontal
When the launch is horizontal (like in graph 2) the initial vertical velocity is 0ms-1 which can be useful for calculating quantities such as the height an object falls or the time taken for an object to reach the ground. If the time can be calculated it is then easy to calculate the horizontal distance travelled by the object.
At an Angle to the Horizontal
When an object is launched at an angle to the horizontal we have both an initial horizontal velocity and an initial vertical velocity. These velocities can be found by resolving the velocity as shown in graph 3.
The horizontal velocity remains the same throughout the motion (as is doesn’t experience any acceleration).
The vertical velocity decreases until the object reaches its maximum height. This is because the acceleration due to gravity is in the opposite direction to the objects vertical velocity.
At the maximum height the vertical velocity of the object is 0ms-1. However the acceleration due to gravity still acts on the object and remains constant.
After the object has reached its maximum height it begins to fall back towards the ground. Now the vertical velocity of the object is in the same direction as the acceleration and so the object speeds up until it hits the ground.
Worked Examples
For all of the examples assume that g = 9.81 ms-2 and ignore the effects of air resistance.
Example 1
A ball is thrown horizontally off a 50m cliff with an initial velocity of 7ms-1. Calculate how long it takes for the ball to hit the ground and how far it will travel in the horizontal direction.
Example 2
A ball is kicked from the ground with an initial velocity of 20ms-1 at an angle of 25° to the horizontal. Calculate the maximum height the ball reaches, the time it takes to fall back to the ground and the total horizontal distance it travels.
Example 3
An archer releases her arrow horizontally along a line passing through the centre of the target which is 10m away from where she is stood. The arrow hits the target 0.25m below the centre of the target. Calculate the time taken for the arrow to hit the target and the horizontal speed of projection.
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