Calculating Final Speed of Engine-less Space Capsule Launched from Earth

[r_swayze]

You launch an engine-less space capsule from the surface of the Earth and it travels into space until it experiences essentially zero gravitational force from the Earth. The initial speed of the capsule is 18,500 m/s. What is its final speed? Assume no significant gravitational influence from other solar system bodies. The Earth's mass is 5.97×10^24 kg, and its radius is 6.38×10^6 m.

I have no idea where to start on this problem. Any help?

 

[tiny-tim]

Hi r_swayze!

Conservation of energy.

 

[r_swayze]

but doesn't the total energy increase as the radius increases?

 

[tiny-tim]

No, total energy remains constant …

why would it not do so?​

 

[r_swayze]

the textbook says it does:

"The total energy of a satellite increases with the radius (in the case of circular orbits) or the semimajor axis (in the case of elliptical orbits). Moving a satellite into a larger orbit requires energy; the source of that energy for a satellite might be the chemical energy present in its rocket fuel."

And don't I need the mass of the satellite to use the energy equations? mass is not given here.

 

[tiny-tim]

the textbook says it does:

"The total energy of a satellite increases with the radius (in the case of circular orbits) or the semimajor axis (in the case of elliptical orbits). Moving a satellite into a larger orbit requires energy; the source of that energy for a satellite might be the chemical energy present in its rocket fuel."

ah … they're talking about the total energy for an orbit.

It stays constant throughout the orbit, but of course is different for different orbits.

Although it only talks about circular and elliptical orbits, the same applies to hyperbolic ones (though of course as a matter of English rather than physics, we would call them trajectories rather than orbits ).

In this question, the capsule is following a single orbit (hyperbolic trajectory), and its total energy stays constant throughout.

(incidentally, even a falling object is following an orbit … one that is so elliptical it's just a straight line that goes back and forward though the centre of the Earth )

And don't I need the mass of the satellite to use the energy equations? mass is not given here.

No, just call the mass m … you'll find it cancels out in the end.

 

[r_swayze]

ok, but then how would I find the radius needed for that orbit? I can't just plug in 0 for F = GMm / r^2, right?

 

[tiny-tim]

ok, but then how would I find the radius needed for that orbit? I can't just plug in 0 for F = GMm / r^2, right?

Stop talking about radius!

"radial distance", ok, or just "distance" or "r"

In the question, "until it experiences essentially zero gravitational force from the Earth" means "at r = ∞" … use that.