Why Might a Space Probe with High Launch Energy Not Escape Earth's Gravity?

[Peter G.]

Hi,

A space probe is launched from the equator in the direction of the north pole of the Earth. During launch the energy given to the probe of mass m is

E=(3GMm/R_E)

Deduce that the Space probe will not be able to travel into deep space.

That is a past paper question my teacher gave me. I am very confused because I was confident it would be able to escape due to the fact the energy is three time that required:

GMm/R_E

Anyone has any explanations or is the question just wrong?

Thanks!

 

[Steely Dan]

Nevermind, looks like the question was mis-typed.

 

[gneill]

Hi,

A space probe is launched from the equator in the direction of the north pole of the Earth. During launch the energy given to the probe of mass m is

E=(3GMm/R_E)

Deduce that the Space probe will not be able to travel into deep space.

If the north pole in question happens to be the geographic north pole, then the probe would have to travel through the body of the Earth to get there!

The question is a bit vague in that it doesn't specify whether the energy imparted is the total mechanical energy with respect to the center of the Earth, the kinetic energy w.r.t. to the launch point frame of reference, or whether it includes the contribution of the Earth's rotation at the point of launch.

If it's the kinetic energy imparted to the probe by all sources with respect to the center of the Earth then subtract the gravitational potential energy and see if the result is negative, zero, or positive and declare accordingly.

EDIT: It occurs to me that the answer to the question may depend upon the definition of the term "deep space". If it implies escape from the Solar System itself, then you'll need to consider another major gravitational potential...

 

[Peter G.]

Looked at the marking scheme and they use a (3GMm/4R_E) so I guess it was a missprint. Thanks everyone nonetheless!

 

[asteriatic]

Hello,

In order for a space probe to escape the gravitational pull of a planet, it must reach a certain velocity known as the escape velocity. This velocity is dependent on the mass of the planet, the mass of the object, and the distance between them. The equation you have provided, E=(3GMm/RE), is the total energy of the space probe at launch, where G is the gravitational constant, M is the mass of the Earth, m is the mass of the probe, and RE is the radius of the Earth.

However, this equation does not take into account the velocity of the probe. In order to escape the Earth's gravitational pull, the probe must also reach a certain minimum velocity, known as the escape velocity, which is given by the equation:

Ve = √(2GM/RE)

Where Ve is the escape velocity. As you can see, this equation does not have the factor of 3 that is present in the energy equation you have provided. Therefore, while the energy given to the probe may be three times the required amount, it does not guarantee that the probe will reach the necessary velocity to escape the Earth's gravitational pull.

Additionally, the direction of the launch (from the equator towards the north pole) may also affect the probe's ability to escape. If the probe is not launched at the correct angle and velocity, it may not have enough momentum to overcome the Earth's gravity and continue into deep space.

In conclusion, the energy given to the probe may be sufficient, but it is not the only factor in determining whether the probe can escape the Earth's gravitational pull. The velocity and direction of the launch are also crucial factors to consider. I hope this helps clarify the confusion.