Projectiles

Projectiles

Last time we were exploring the world of the angry birds, and now I want to look a bit more at how we know how far something will go when it’s thrown. As children we learned this by trial and error, improving our throwing technique the more we played catch, but this isn’t a great method if you want to fire something more important onto a specific target!

This is where the maths comes in, specifically the equation of motion (sometimes called the SUVAT equation), and it tells us the relationship between speed (or to be more precise velocity, which is the same as speed but it also takes direction into account, so whereas with speed you might say a car was reversing at 5mph, you could also say the velocity was -5mph), acceleration and time.

This equation can take three forms, so it’s important to see what bits of the problem you have, and what you need to find out, to make sure you use the right one

Here is where the notation can get a little tricky, and it’s all about making sure you know which symbols are which, otherwise it’s impossible to get an answer.

a=acceleration
u=starting velocity
v=ending velocity
t=time taken
s=distance travelled

So, we want to know how far we can get our projectile to go, and then see how we can get it to go the furthest possible distance. We know that we have a starting velocity that depends on our catapult, that at the end our projectile will be stopped, and that once in motion the only acceleration acting on our projectile is gravity, which we’ll say is 10ms^-2.

To summarise we have u, v, a and we need t so that we can find s, so we’ll need to use equation 1 to get the time.

It’s clear to see that we’ve got a little difficulty with our initial velocity u, it’s pointing in both the x and y directions, which makes things a little confusing, because our motion equations want us to be travelling only in the x-direction or only in the y-direction. But never fear, we can use trigonometry to resolve this issue, by calculating how we should divide up our velocity u.

So we can see in the y-direction, the proportion of the velocity u that belongs to the y-equations is u sinA where A is the angle from the horizontal in degrees. Writing out our equation for the y-direction then we get

because we know that the acceleration a in the y-direction is gravity, 10ms^-2

Now this all looks a little bit horrible, but if you work through it it’s not too bad. Let’s work out an example. Let’s say the angle is 30º, and the initial velocity is 5ms^-1

If we want to work out the best angle to get the maximum distance then we just need to keep doing the sum, with u staying the same, and the angle A changing. Repeating the same sum to find the best answer is called ‘trial and error’ and it’s a great way to find things out like this.

Let’s finish with a table, with the angles we want to try out, and then we can work out the distances the projectile will go.

Angle A (degrees)	Distance s (m)
10	0.43
20	0.80
30	1.08
40	1.23
45	1.25
50	1.23
60	1.08
70	0.80
80	0.43

So the best angle for getting your projectile the furthest is 45ºC, now go get those pigs!