Difficult Astronomy question

I have to calculate how fast a planet would have to rotate in order to have a negative escape velocity.

What's the method behind working out the answer?

Escape velocity is dependant on the mass of the body, which determines its gravitational pull which in turn determines the speed it would take to overcome the pull of gravity based on that mass to escape from the body. To have negative escape velocity would mean that there is no mass, or gravity, and in fact the pull of gravity is in the negative range, or less than zero. Even the moon has an escape velocity, which is why astronauts had to use fuel to escape from its surface. A body with no mass doesn't exist. I don't believe that such a planet is possible, because negative escape velocity means negative energy used..lower than no energy used. Unless you have some sort of antimatter parallel universe, there is no such planet.

I think Jim's question relates to the fact that rockets are not simply launched from the surface of the Earth. They actually use the spin of the planet to help toss the rocket off the Earth.
In theory the faster the spin the less added push the rocket engine would need to achieve escape.

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I think I get what you are trying to ask, and if so, there is no generic answer - it all depends on the details.

The escape velocity from a planet depends on its mass, and the distance from the center of mass. I imagine we are talking here about the planet;s surface, in which case for 'distance from the centre of mass' you can pretty much read 'equatorial radius'. (The inclusion of the word 'equatorial' is very important here, because a fast-spinning planet would be far from spherical.)

OK - so now you have the mass and equatorial radius, you can calculate escape velocity at the equator. In principle, it is possible that the planet could rotate fast enough that, at the equator, the surface was moving at that speed, so that anything 'falling' off the surface would be ejected into space. Of course, in reality that couldn't happen, because the planet could not possily e rigid enough to hold together until anything like that rotational velocity.

Anyway, once you have calculated the escape velocity, you can calculate the rotation rate easily enough, because you can calculate the circumference from the radius (which you already know). Then (for example) divide the escape speed in km/hr by the circumference in km, and you have to rotation rate in revolutions/hr.

Escape velocity has nothing to do with a planet's rate of rotation. It is more to do with the amount of mass and how far away from the center you are.

Also, there is such thing as a "negative" velocity.

"Calculate"?

Do you want a general formula or a specific example?