STEP I 1997 question 11 solution

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Change in kinetic energy = work done on the particle:

Differentiating:

One particular solution to the equation is . The corresponding homogeneous equation

has solutions

Thus the general solution is

At we have :

Thus

as required.

The maximum height is when :

.

On the downward path, gravity is increasing the particle's kinetic energy instead, and we get the differential equation

which clearly has a solution that's different from the given equation. To see this, the general solution is

instead. We have that at

.

and so

and

thus the difference from when the particle was heading upwards is only that there is a plus sign instead of a minus signe before the '1'. But nevertheless it's different and so the equation (*) does not hold on the downward path.

Solution by ukgea