An astronaut shipwrecked on a distant planet with unknown characteristics is on

Question

An astronaut shipwrecked on a distant planet with unknown characteristics is on top of a cliff, which she wishes to descend. She does not know the acceleration due to gravity on the planet, and she has only a good watch with which to make measurements. She wants to learn the height of the cliff, and to do this she makes two measurements. First, she lets the rock fall from rest off the cliff edge; she finds that the rock takes 4.15 s to reach the distant ground below the cliff. Second, she releases the rock from the same spot but tosses it upward so that it rises 2 m before falling to the distant ground below the cliff. This time the rock takes 6.30 s to reach the ground. What is the height of the cliff.

Explanation / Answer

t^2 = 2h/g.

Let x be the time for the stone to fall through 2m, from the highest point to the cliff. x^2 = 4/g.
Let y be the time for the stone to fall to the ground from the highest point.
y^2 = 2(H + 2)/g = 2H/g + 4/g
Let z be the time for the freely falling stone to reach the ground from the cliff.
z^2 = 2H/g

from these three equations :

y^2 -x^2 = z^2, implies [y + x]*[y

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