Question

An astronaut shipwrecked on a distant planet is on top of a sheer vertical cliff (let’s call it point P), which he wishes somehow to climb down. He does not know the acceleration due to gravity on the planet, and he has only a good watch with which to make measurements. He wants to learn the height of the cliff (from P down to the bottom). To do this,he makes two measurements. 1) He drops a rock fall from rest off the cliff edge (from point P) and finds that the rock takes 4.15 s to reach the bottom. 2) He tosses another rock straight upward from P such that it rises a height of what he estimates to be 2 m (up from P) before it falls to the ground below. This time the rock takes 6.30 s from the original toss to hit the bottom. What is the height of the cliff (point P down to the bottom below)?

Answer 1

We have the equation of motion,

                             

Where, S is displacement

             u is initial velocity

            a is acceleration

            t is time of motion

Motion of the rock when astronaut directly drop it fromm the P:

                      u = 0 m/s

                            t = 4.15 s

Let acceleration due to gravity of the planet is g' and height of the cliff is h. Therefore, the acceleration of the rock will be g'.

Applying the equation of motion, we get

                              

                       or,                                        ................................(1)

Motion of the rock when astronaut tossed it from the P:

We know that at the topmost position, the velocity of the rock will be zero.

Downward motion from the topmost position:

We can write, u = 0 m/s (at the top )

                        a = g'

                        S = h+2 (given the rock travells 2 m in upward direction)

Applying the equation of motion, we get

                       

              or,                                    ...............................(2)

Putting the value of g' from equation (1) in (2), we can write

                      

                 or,

                 or,

Hence, the height of the cliff is 1.53 meters.

For any doubt please comment.