# Converting energy to impulse

1. Aug 17, 2013

### BitWiz

Problem: A nuclear generator produces energy used to accelerate a rocket. Given energy and mass, what is the acceleration a?

Nuclear generators produce energy in GeV or joules which have the dimensions L-2T-2M which has a force-over-distance (MaL) component which seems inappropriate or at least messy for acceleration. A better(?) unit, especially for delta-v, is force-over-time, ie impulse, with dimensions LT-2TM or LTM (momentum).

Impulse is then joules divided by (some) velocity. I can convert(?) by dividing joules by a "universal velocity" (the particle guys use c), or by forcing this (artificial) velocity to 1.0.(?)

In Case 1), a post on Wikipedia (here) in the "Momentum" section asserts that 1.0 GeV = 5.3 x 10-19 kgm/s which seems small by several orders of magnitude -- perhaps I don't understand what these units are really meant for ...

In Case 2), I'm forcing delta-v = aT = joules (where force is applied over exactly 1 meter) by using a unity velocity.

Despite the dimensional contortions, does the latter work? If so, are there hidden implications? For instance, the reaction mass used to propel the rocket could only be accelerated over 1.0 meters or this relationship falls apart, correct?

Are there any generalized equations for energy and power => impulse/momentum?

Chris

2. Aug 18, 2013

### BitWiz

Edit: Screwed up a superscript above. Should be "impulse = dimensions LT-2TM or LT-1M (momentum)."

Chris

3. Aug 21, 2013

### eigenperson

By itself, the energy released by the rocket engine is not enough information to tell you what the impulse will be.

This can be easily seen in the following comparison:

In one second, Rocket 1 produces 4 g of exhaust at 500 m/s.
In one second, Rocket 2 produces 1 g of exhaust at 1000 m/s.

Assume both rockets weigh many tons.

These rockets put the same amount of energy into their exhaust. But the impulse of Rocket 1 is 2 times greater.

Now, this is not really a proper analysis because we didn't consider how much of the energy went into the rocket itself, but because the rockets weigh many tons (and thus achieve only very low speeds over the time period in question), the amount of energy in the rocket is negligible.

***

OK, so what do you need, in addition to the energy, to find out the impulse? Well, it's enough to know the exhaust velocity or the exhaust mass. (More properly, since rockets produce exhaust over time, you should know the power and the exhaust mass flow rate, but we'll let that go by assuming the time period is infinitesimal.)

If you know the exhaust velocity and the exhaust energy is E = (1/2)mv2, then the momentum of the exhaust is mv = 2E/v. Alternatively, if you know the exhaust mass, then the exhaust velocity can be calculated as v = (2E/m)1/2, and then mv = 2E/v as above.

Obviously the momentum of the rocket is equal and opposite to the momentum of the exhaust.