Can a rocket go faster than the speed of its exhaust?

Oh, hey, you know what? This question might be more interesting than it looks. The most obvious answer would be no, it can't go faster than its exhaust. BUT, it's not just the velocity, but the acceleration which becomes the important factor here.

Suppose the exhaust is clocked at 1 km/sec. The ship will get up to that speed, but what then? The speed of the exhaust is relative to the space ship and the ship is accelerating. According to relativity, it would be like starting from zero again. So the fuel may have a certain velocity, but applied over time results in an acceleration which is theoretically unlimited.

In practice, it would be limited, again because of relativity. As you reach relativistic speeds, the amount of energy required increases exponentially and the amount of acceleration possible eventually reaches zero as you approach the speed of light.

Wow, as fundamental as that seems, I never considered it! Thanks.

Steve, it does involve relativity in two respects. First, because the velocity of the exhaust is relative to the space ship, not to some fixed starting point. Second, at the theoretical extreme, where it must be conceded that the speed of light couldn't be reached. IOW, the ability to continue accelerating is not really unlimited.

When I talk to schools about how rocket and jet engines work, I compare it to a man standing on a cart with 4 wheels, like a small railroad handcar. He throws a bowling ball off the back of the cart, and the cart moves forward slightly. He continues to throw bowling balls off the back, and the cart continues to accelerate. The bowling balls are not pushing against anything, and it does not matter how fast they are moving with respect to the ground. What matters is that they are transferring momentum to the cart when they are tossed off the back.

A rocket has a limited supply of bowling balls, but the ones it has are "big ones." A jet engine can keep pushing bowling balls out the back as long as it can keep pulling air in the front and mixing it with fuel.

ADDED: Steve,

You were correct to start with. The velocity of the exhaust relative to the velocity of the vehicle through the air - or the lack of air - makes no difference whatsoever. The thing that matters is the momentum of the exhaust vs the momentum of the vehicle in the vehicle reference frame. The exhaust does not "push against" anything (ADDED: at least nothing behind the vehicle), but that is probably the original misconception .

ADDED(2) I don't think I made this clear. A rocket free of gravity will accelerate in accordance with delta mv (exhaust) = -delta mv (vehicle). The initial velocity really doesn't matter, and might be difficult to specify anyway, depending on how far you are from a planet that you can refer to. The point is, that's how the rocket works in the atmosphere as well. The exhaust does not care how fast the air is whizzing past, it only cares that it is being blown aft from the nozzle, creating a huge momentum transfer to the vehicle.

Brant nailed this one!

What is this, quiz night for the regulars here in Astronomy & Space? If so, then I like it!

Just to throw my own two cents into the discussion:

All a rocket engine is doing is taking momentum from the rocket, and giving it to the propellant in a way that removes the propellant from the system. So if the rocket is going at 500k km/sec, and the exhaust velocity is 100km/sec, at the end, you'll have exhaust particles going at 499,900km/sec, and a rocket going just a little faster in an amount equal in momentum to the loss in momentum of the exhaust particles.

As long as momentum is conserved, Newton's not going to get you for violating law #3.

(Meaning that you are still obeying it, so he can't get you on that)

It depends on the mass of the reaction mass. If I throw a 100-ton steel ball in space, I will actually be throwing myself instead, and much faster than the steel ball. Rocket exhaust typically goes faster than the rocket, however, since high-speed but low-mass reaction mass is preferable. That's because high-mass reaction masses can't leave a gravitational field very well, nor does it allow for easy course corrections.

A little bit Plus acceration"

Uhm, probably from newton's third law.