# Cannon Balls

How high will a ball taking a million seconds to fall travel?

## Problem

A cannon ball is fired vertically upwards into the air. How fast would it have to be fired to take 1 second to land?

How fast would it have to be fired to take 10, 100, 1,000 or 1,000,000 seconds to land?

What would be the highest point of the ball in each case?

(Assume that gravity is a constant 10ms$^{-2}$ in your calculations.)

Given that the radius of the earth is about 6000km, which of your calculations would give a good approximation to reality? At what speed would the approximation break down, in your opinion?

Extension activity: Suppose that the balls are fired upwards from a trampoline with coefficient of restitution 0.5. In each case, after how many bounces would the balls bounce less than 1m high? Try to make an estimate before performing a full calculation.

Extension problem: why not try the extension question Escape From Planet Earth?

## Getting Started

What equation do you need to solve?

## Student Solutions

We recieved solutions from Vladimir Radmanikov from South Island School, Samuel from Manor Park Community college and Laura from Millhaven school, who said the following:

If we take upwards as positive $x$ then this is motion under a constant downwards acceleration $a=-9.8$. We need to use the formula

$$

x(t) = v(0)t+\frac{1}{2}at^2= v(0)t-4.9t^2\;.

$$

We want to know the starting speed $v(0)$ to make it hit the ground $x=0$ in $1\mathrm{s}$.

So, we must solve

$$

0 = v(0)\times 1 -4.9\times 1^2\;.

$$

So, the ball must be shot up at $4.9\mathrm{ms}^{-1}.$

For other times we have

$$

v(t) = 4.9t\;.

$$

So, double the time requires double the speed and so on. So, $10\mathrm{s}$ travel time requires $49\mathrm{ms}^{-1}$ etc., as in this table

Travel time | $10\mathrm{s}$ | $100\mathrm{s}$ | $1000\mathrm{s}$ | $100,000\mathrm{s}$ |

Speed | $49\mathrm{ms}^{-1}$ | $490\mathrm{ms}^{-1}$ | $4.9\mathrm{kms}^{-1}$ | $490\mathrm{kms}^{-1}$ |

Clearly the last one will go straight into space!

You used various method to find the higest point of the cannon ball, which occurred half way through the journey.

Samuel said

We need to work out when the ball stops moving to find the highest point each time. Differentiating the first equation gives us

$$

v(t) = v(0) -9.8 t\;.

$$

If $T$ is the time of maximum height then setting $v(T)=0$ gives $T = v(0)/9.8= t/2$. So it reaches the maximum height half way through the travel time in each case. Putting $t=5,50,500,500000$ into the first equation gives

Travel time | $10\mathrm{s}$ | $100\mathrm{s}$ | $1000\mathrm{s}$ | $100,000\mathrm{s}$ |

Maximum height | $122.5\mathrm{m}$ | $12.25\mathrm{km}$ | $1225\mathrm{km}$ | $12250000\mathrm{km}$ |

Laura noted

By conservation of energy the cannon ball must hit the ground with the same speed as it left the ground, so the up and down parts of the journey are mirror images. So the maximum height is reached exactly half way through the journey. So, at times $5\mathrm{s}$, $50\mathrm{s}$, $500\mathrm{s}$, $50000\mathrm{s}$.

Vladimir made use of another form of the kinematic equations

To find the maximum height I can use this formula

$$v(t)^2 = v(0)^2 + 2ax(t)\;.$$

At the highest point $v(t)=0$, so

$$

x_{max} =\frac{v(0)^2}{2\times 9.8}\;.

$$

This method also gives the values in the previous table.

Vladimir though that, since the earths radius is $6000\mathrm{km}$ the cannon ball taking $1000\mathrm{s}$ was likely to return to earth, as this has a maximum height of $1225\mathrm{km}$, and was therefore still in range of gravity (even taking the decay of gravity into account). Laura noted that the escape velocity from earth is around $11\textrm{ km per second}$, so all but the final cannon ball would return to earth.

Well done to you all!

## Teachers' Resources

### Why do this problem?

This short problem is a reasonably routine application of kinematics; the interest lies in the numbers obtained and the questions concerning the validity of a physical model of constant gravitational force.### Possible approach

Students could be asked to make an estimate of the speeds and heights before starting the calculation. Developing a skill and habit for estimation is very useful in more advanced applications of mathematics.### Key questions

- What has the radius of the earth got to do with this problem?

### Possible extension

Try the follow up problem Escape
from planet earth .

You could also extend this
to suppose that the balls are fired upwards on a
trampoline with coefficient of restituion 0.5. How many bounces
would it take for each ball to bounce less than 1m high?