### A Close Match

Can you massage the parameters of these curves to make them match as closely as possible?

### Prime Counter

A short challenge concerning prime numbers.

### The Right Volume

Can you rotate a curve to make a volume of 1?

# Crazy Cannons

##### Age 16 to 18 ShortChallenge Level

Two cannons both fire balls at speed $100 \mathrm{ms}^{-1}$. One of the cannons is fixed at an angle of $45^\circ$ to the horizontal and the other is fixed at angle $30^\circ$ to the horizontal. The cannons are set up facing each other at a distance $D$ from each other. One cannon is fired and then, $T$ seconds later, the other cannon fired.

The balls subsequently strike each other in mid air.

Show that that $T$ must lie within a range of values and that both $T$ and $D$ must be greater than zero.
Did you know ... ?

The mathematics used in this question, along with an understanding of how gravity works at large distances, is sufficiently complex to send rockets to the moon and the planets of our solar system.