Mechanical Energy and Frames of Reference

[CharlesEster6]

Hello,

I've come up with a simple, imaginary situation that bugs me. Could someone help resolve my confusion?

Let's say that a spaceship is traveling through space, and two people- observer 1 and observer 2- are watching it. Both observers can monitor the ship's speed and the amount of fuel in its "gas tank." Initially, the ship is moving at speed v with respect to observer 1, and is at rest with respect to observer 2.

Now the spaceship accelerates, so it is moving at speed v with respect to observer 2 and speed 2*v with respect to observer 1. Observer 2 notes that 0.5*m*v² (m is the mass of the ship) joules of energy are withdrawn from the fuel during this period of acceleration, but to observer 1 it appears that 0.5*m*((2v)² - v²), or 0.5*m*3v² joules of energy are consumed. It can't be true that observer 1 records three times more fuel used than observer 2; what accounts for this "inconsistency?"

Thank you!

 

[peteratcam]

You need to take into account the energy of the burnt fuel ejected. The energy from burning the fuel goes into ejecting hot gases in one direction and moving the spaceship forward in the other direction in such a way that the total momentum is conserved. If you are careful about taking into account the energy of the gases, there is no inconsistency.

A similar problem is the following:
A ball is stationary at the top of a hill, and someone gives it an infintesimal knock and it rolls down the hill losing potential energy, and gaining kinetic energy until it has final velocity v. (No friction). This sounds sensible.

Now imagine the situation from an observer moving at velocity v. He sees the ball moving away from him, then it rolls down the hill losing potential energy, and eventually comes to rest, losing kinetic energy too! This sounds crazy.

The resolution is along the same lines as your spaceship.
Peter

 

[Dale]

Hi CharlesEster6 and peteratcam, Welcome to PF,

This is a good question, and peter is correct about the resolution. There is a https://www.physicsforums.com/showthread.php?t=199087" thread about this same subject where the math in the tutorial was discussed in detail.