Challenge Level

A particle is projected with speed $10 \mathrm{m s}^{-1}$ from a flat horizontal surface. Find, with proof, the angle from which it should be projected to maximise the distance travelled before it hits the surface. Does this angle depend on the speed of projection?

The particle is now projected with speed $10 \mathrm{m s}^{-1}$ from a height of $2$ metres.

From what angle (to 3sf) should it now be projected to maximise the distance travelled before it hits the surface? Does this angle depend on the speed of projection?

Did you know ... ?

The modelling assumptions of constant gravitational field and no friction opposing motion are good ones, leading to simple equations which always have parabolas for solution. Once these modelling assumptions are, rightly, challenged, the resulting equations become 'non-linear' and very difficult to solve. Mathematicians often take the parabola as a starting point to solving the more complicated equations and vary the solution a little to try to fit it back into the new equations. You can see an aspect of this process in the solution to this problem.

The modelling assumptions of constant gravitational field and no friction opposing motion are good ones, leading to simple equations which always have parabolas for solution. Once these modelling assumptions are, rightly, challenged, the resulting equations become 'non-linear' and very difficult to solve. Mathematicians often take the parabola as a starting point to solving the more complicated equations and vary the solution a little to try to fit it back into the new equations. You can see an aspect of this process in the solution to this problem.