Kinetic energy of space probe launched from planet

[1MileCrash]

Homework Statement

Zero, a hypothetical planet, has a mass of 3.0 x 10^23 kg, a radius of 3.0 x 10^6 m, and no atmosphere. A 17 kg space probe is to be launched vertically from its surface.

If the probe is launched with an initial kinetic energy of 5.0 x 10^7 J, what will be its kinetic energy when it is 4.0 x 10^6 m from the center of Zero?

Homework Equations

The Attempt at a Solution

I figure that mechanical energy is conserved, and so

K1 + U1 = K2 + U2

With U given by GmM/r.

So

[itex]K_{2} = K_{1} + GmM(r^{-1}_{2} - r^{-1}_{1})[/itex]

With r1 = 3.0x10^6, and r2 = 3.0x10^6 + 4.0x10^6 = 7.0x10^6

Which gives me the negative value -14794285.71 for K2.

I have no idea what is wrong with the equation I'm using.

Any ideas?

 

[SHISHKABOB]

gravitational potential energy is negative

also I think that your r² is a bit too big

 

[1MileCrash]

Yes, so

K1 - (GmM/r1) = K2 - (GmM/r2)

And

K2 = K1 - (GmM/r1) + (GmM/r2)
K2 = K1 + (GmM/r2) - (GmM/r1)
K2 = K1 + GMm((1/r2)-(1/r1))

Which is what I used..

 

[SHISHKABOB]

Yes, so

K1 - (GmM/r1) = K2 - (GmM/r2)

And

K2 = K1 - (GmM/r1) + (GmM/r2)
K2 = K1 + (GmM/r2) - (GmM/r1)
K2 = K1 + GMm((1/r2)-(1/r1))

Which is what I used..

I did

K2 = K1 - U1 + U2
K2 = K1 -(U1 - U2)
K2 = K1 -GmM((1/r1) - (1/r2))

and got 21652500 J

I also got the same number when I used your method

I think the problem is your value for r2

 

[asteriatic]

I would like to point out that the initial kinetic energy of the space probe does not depend solely on the mass and radius of the planet, but also on the velocity at which it is launched. Therefore, without knowing the initial velocity of the probe, it is not possible to accurately calculate its kinetic energy at a specific distance from the center of the planet.

However, assuming that the probe is launched with a velocity that allows it to escape the gravitational pull of Zero, the equation for conservation of mechanical energy would be valid. In this case, the initial kinetic energy of the probe would be equal to the gravitational potential energy at the surface of the planet, which can be calculated using the given mass and radius of Zero. The final kinetic energy at a distance of 4.0 x 10^6 m from the center of Zero can then be calculated using the same equation, as you have done.

It is possible that the negative value you obtained for K2 is due to a mistake in the calculation or a sign error. It would be helpful to double check your calculations and make sure that you are using the correct units. Also, keep in mind that the final kinetic energy may be negative if the probe is slowing down due to the gravitational pull of Zero.