velocity and height of hill?

Question

Coasting from rest down a certain hill, whose slope is variable, I reach a speed of 18.02 m/s at the bottom. If I coast from rest down the second half of the hill I reach a speed of 15 m/s. Ignoring the effects of air resistance and friction:

How fast would I therefore be going if I coasted from rest down the first half of the hill?
How high would I have to climb from the halfway point reach the top?

Answer 1

If u is the speed at the end of the first half and s is the vertical distance traveled during the second half,

Using the equation v^2 - u^2 = 2as, we have,

18.02^2 - u^2 = 2gs = 15^2 - 0^2. where s is the vertical distance of the second half.

u^2 = 18.02^2 - 15^2

u = 9.98m/s
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Since at the end of first half the speed is u,

The vertical distance H of the first half is

u^2 - 0^2 = 2 g H

H = 9.98^2 / [2x9.8] = 5.08 m.
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Answer 2

First use the velocity relation
v = a * t where a=9.8 m/sec^2 to find the time to reach the speeds involved.

Then, use the position relation
x = (a/2) * t^2 to find the positions corresponding to the times.