How does $p=fv$ hold when $v=0$ and an engine is working?

Question

Standard high school equations:

Work  = Force x Distance
Power = Work / Time
      = Force x Distance / Time
      = Force x Velocity

When a space rocket takes off its velocity is initially zero, this equation tells us that the power of the engine is also zero. Looking at all the flames coming out the bottom, this doesn't make intuitive sense.

Consider also a rocket in free space which fires its engines at the front until it reverses direction. This tells us that the power of the engine starts out negative and passes through zero as it tends to infinity. Again, this doesn't make intuitive sense.

What's wrong with my intuition?

Answer 1

In your first example, a rocket on a launchpad that has just started its engines, the equation is exactly correct. The motor is generating huge force, but it has not yet moved the vehicle, so no work or power is being generated. In reality, that moment of no work or power lasts only a tiny fraction of a second. The power moves the rocket and then you can measure/calculate it. And you have forgotten that the rocket may be still, but the rocket gases are moving really fast out of the bottom of the rocket. How much power does that represent?

In your second example, a rocket decelerating, you are touching on the subject of Calculus! Here, the question is one of change of velocity as deceleration. What you have touched on is the same question that plagued Newton, if velocity changes over time, how do I calculate exactly where the rocket will be and how fast it is traveling at that moment.

Answer 2

When the rocket sits on the launch pad and the engine fires up, an enormous amount of energy is expended moving the exhaust gases and heating the air. No work is done to move the rocket, but that doesn't mean no energy is expended. But the kinetic / potential energy of the rocket are not changing (actually the potential energy of the rocket becomes less because its mass is going down but that is irrelevant to the argument here).

It's like the old problem of holding a heavy object in your hands. According to your high school physics, you are doing no work since the object is not moving, but clearly you get tired standing there holding it. That's because sometimes there are (chemical, biological) processes involved in generating a force that result in the expenditure of energy even though there is no useful (in the physics sense) work being done on the object.

As for your second question about the rocket in space reversing its thrust: when you slow down an object you can say you are doing negative work on it (you are taking away energy from it). In the case of your rocket, you can see that the exhaust gases will be going faster than your chemical burn is allowing (if the rocket is traveling at 1000 m/s and the exhaust velocity is 4000 m/s, then the gases will be traveling at 5000 m/s relative to space - so clearly the rocket did work on the gas. Consequently the gas did "negative work" on the rocket.)

I am not sure about your comment about the power tending to infinity... maybe you can clarify what you are struggling with there.

Answer 3

Useful work=0, power DRIVING the engine =0, ALL useful power lost in the form of wasteful energy, example: exhaust gases, heat, sound and all that..