Physics problem dealing with work and energy?

Question

A crate of mass 100 kg sits on top of cliff 50 m high. The crate is then pushed off of the cliff and falls straight down until it hits the ground. Assuming negligible air resistance, determine (a) the potential energy of the crate and (b) the final velocity of the crate upon reaching the ground.

(a) P = mgh = 4.9x10^4 J

(b) mgh = (1/2)mv^2 > v = square root (2gh) = 31 m/s

my question is, is this correct?

also, the potential energy is equal to the kinetic energy and the work done by gravity right?

Answer 1

I presume you mean the potential energy atop the cliff which is mgh=100kgx9.8m/s/sx50m=49,000J (rather than potential energy on the ground).  If so, your answer is correct.

Neglecting friction, the total energy is always the same.  At the top, there is no KE and all the energy is in PE; at the bottom all the energy is in KE, so we can equate PE at the top with KE just before impact and write:

mgh=1/2 mv^2 or v=sqrt[2gh]=31.3 m/s

the total energy is constant; the sum of PE+KE at any level is the same; PE is not always equal to KE, but it is correct to say that the PE at the top is equal to the KE at the bottom

the change in KE is equal to the work done by gravity; here the change in KE is equal to the change in PE which is the same as the PE at the top since there is no PE at the bottom

Answer 2

You are on the right track, but I think you have somethings slightly confused. You got the potential enegy calculated right for when the box is a the top of the cliff. As it falls that potential energy is converted into kinetic energy. So when it hits the bottom, it still has the same total energy (conservation of energy), but that energy is now all kinetic. So you can say that KE = 0.5mv^2.

Since you already calculated how much energy it had at the top, you know how much energy it has at the bottom.

Your answer for both parts is right, but I your final sentence made me think you may be a bit confused about what is happening. It is not true that PE = KE, because you can convert one type of energy into another. However PE + KE = constant.