STEP III 2007 question 10 solution

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Draw the obligatory diagram, and we see that in the (x,y) coord frame, acceleration due to gravity is .

Thus the equations for x and y are:

When the particle hits the ground, y=0 (and ), so . [B](1)[/B]

The rebound only affects the y component of the velocity, so if it is to retrace its path, we must have and so . [b](2)[/b]

Divide (2) by (1) to get and so [b](3)[/b].

Rewrite (2) to get at time of landing. Substitute into the equation for x to get the maximum range:

as required.

So then

.

(Using (3) to tell us ).

as required.

Now

, with equality when .

Thus with equality when .

Then with equality when .

Solution by DFranklin.