- Recap
- Projecting a particle directly upwards
**Projecting a particle at any angle**- Projectile motion formulae

# Part 3: Projecting a particle at any angle

## The key idea

In this section, the projectiles are under the influence of gravity alone—i.e. the only relevant force is weight. This means that projectiles experience a constant downward acceleration of \(g \text{ ms}^{-2}\), and zero acceleration in the horiztontal direction. This means that horizontal velocity remains constant during the projectile’s flight.

This applet allows you to view the trajectory of a projectile, P, in 3D from three perspectives. The “side-on” view shows the plane containing the two given axes, and this plain contains the trajectory of the projectile. Viewing the trajectory from the point of projection, the trajectory appears exactly like that of a projectile thrown straight up. Viewing from above, we see that horizontal velocity remains constant.

Once the animation has completed, you can choose to highlight the position of P at equally spaced time intervals. When viewing these, click on any of the position markers to see how their horizontal spacing is constant, and click again to see that their vertical spacing clearly is not constant.

## Displacement-time and velocity-time graphs for a trajectory

This applet features two panes. The graph in the left-hand pane shows the trajectory of a projectile, with the axes showing the horizontal and vertical displacements in metres.

The right-hand pane can show one of four graphs relating to the projectile:

- Horizontal displacement against time
- Vertical displacement against time
- Horizontal velocity against time
- Vertical velocity against time

To cycle between these four graphs, simply **click the label on the vertical axis of the right-hand graph**. When displaying a velocity time-graph on the right, vectors showing the horizontal and vertical components of the projectile’s velocity will be shown on the projectile in the left-hand pane.

You can animate the projectile by clicking the **animate** button. Once started, buttons to pause and resume the animation will appear.

Play with the animation a few times, comparing the trajectory against each of the four graphs. Everything should be consistent with the big idea above, and your prior knowledge of displacement-time and velocity-time graphs. Check with your teacher if you feel something doesn’t make sense.

## How the initial speed and angle affect the trajectory of a projectile

This applet lets you see the range and maximum height of a projectile for different initial speeds and angles of projection.

First pick an initial speed and stick with it. Play with the angle of projection. Which angle results in the greatest range? Which angle results in the greatest height reached?

Now pick an angle less than 90º and stick with that. Play with the initial speed. Note how doubling the speed results in a range that is 4 times longer and a maximum height that is 4 times greater. In general, multiplying the initial speed by \(n\) results in both the range and maximum height being multiplied by \(n^{2}\).

## Solving problems involving projectiles

You should also be able to split a velocity into perpendicular components (most commonly into horizontal and vertical components):