Arocket is continuously firing its engines as it accelerates away from earth. for the first kilometer of its ascent, the mass of fuel ejected is small compared to the mass of the rocket. for this distance, which of the following indicates the changes, if any, in the kinetic energy of the rocket, the gravitational potential energy of the earth-rocket system, and the mechanical energy of the earth-rocket system?
a) ke increases, pe increases, me increases
b) ke increases, pe increases, me constant
c) ke increases, pe decreases, me decreases
d) ke decreases, pe increases, me constant

Option A is correct. See explanation below.

Explanation:

Notice two assumptions in this problem:

1) The rocket is in the first kilometer of its ascent

2) The mass of fuel ejected is small compared to the mass of the rocket

3) The rocket is accelerating (Of course it should be accelerating in its first kilometer)

Now, as the rocket accelerates, it means its speed increases and so does its KE because KE is directly proportional to speed squared.

Potential energy clearly increases because the rocket is on ascent and so is increasing in height. Remember that PE  = mgh and so is directly proportional to height.

Total mechanical energy obviously increases because ME = PE + KE. So if PE and KE are increasing, then ME also increases. Well, some may come up with the argument that Energy is conserved, so ME cannot increase, but recall that fuel is being ejected and combusted. This means chemical energy is used up in the process. It is that chemical energy that changes to Mechanical energy, thereby increasing it.

So yes, the answer is (A).