Mechanical Energy of a Spacecraft

kitz

I have two questions I'm a bit confused on...

1st:

Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was $$2.87 \times 10^{6} km$$ from the earth and traveling at $$1.20 \times 10^{4} km/h$$ relative to the earth.
a.) At this time, what was the spacecraft's kinetic energy relative to the earth?
b.) What was the potential energy of the earth-spacecraft system?

Common sense will tell me that I can find one, once I figure out the other... Now, the first time I approached this, I used the law of conservation of energy... But don't think I'm going the right way in finding the correct initial kinetic and potential energy. For this type of problem, although the terms are different for finding potential energy, is the method to find kinetic energy still the same:

$$K = \displaystyle{\frac{1}{2}}mv^2$$?
If this is true, does this mean that the initial velocity is $$1.20 \times 10^{4} km/h$$?

I understand at 10 days, the orbiter is travelling that fast... But that speed can't be constant, can it? Wouldn't it be decreasing as it escapes earth's gravitational pull?

How would I use the velocity at 10 days to find the kinetic energy? If I simply plug in the given velocity into the kinetic energy formula, I get some insane number that can't be correct...

Any pointers??

2nd:

For a problem,
I found the speed of a spacecraft at which it would crash into earth, assuming it were a distance of $$\infty$$ away, and no other astronomical object's gravitational pull affected it.

$$s_{e}=\sqrt{\displaystyle{\frac{2M_{e}G}{R_{e}}}}$$

And this is what I was then asked:

Now find the spacecraft's speed when its distance from the center of the earth is $$R=\alpha R_{\rm e}$$, where $$\alpha \ge 1$$.
Express the speed in terms of $$s_{e}$$ and $$\alpha$$.

So, this appears to be a simple algebra problem. And after a bout of apparently incorrect algebra, I got

$$s_{\alpha}=\displaystyle{\frac{s_{e}}{\alpha}}$$

Eh... What did I do wrong here??

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Doc Al

Mentor
For the first problem: You are given the speed and distance from the earth at a specific moment in the life of this spacecraft. The speed allows you to calculate the KE; and the distance allows you to calculate the PE of the system. It's that simple. (The stuff about 10 days is just to add a little background to the problem; conservation of energy will not help you answer the questions.)

For the second problem: Yes, you probably made an algebraic error; show what you did and we can take a look. (Hint: Find the speed at $r = \alpha R_e$ and compare it to the speed at $r = R_e$. Divide one expression by the other to get the ratio.)

kitz

Hmm... In using the given speed and plugging it into $$K = \displaystyle{\frac{1}{2}}mv^2$$. And in using that, for KE, I'm getting $$4.53\times 10^{10}$$, which isn't correct... Am I doing the right thing?

For the Potential Energy, using the distance, I'm getting $$-2.71\times 10^{10}$$... Which also isn't correct... What am I doing wrong?

...I'm using $$U=-\displaystyle{\frac{GM_{e}m}{r}}$$
And am adding the distance given to the radius of the earth... $$(6.38 \times 10^{6}m+2.87 \times 10^{6} m)$$

For the second question... In dividing one expression by another, I get:
$$\displaystyle{\frac{s_{e}}{\sqrt{\displaystyle{\frac{2M_{e}G}{R_{e}\alpha}}}}}$$??

And It's supposed to be in terms of $$s_{e}$$ and $$\alpha$$... Am I going about this wrong?

Thank you sir!

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kitz

AHH... for the first question, potential energy part, it was in km, and I was treating the kilometers as meters...

However, for the first part of the question, the speed is in km/h. Should I convert to m/h or m/s? For the KE formula, does it have to be in m/s?

kitz

Doc Al said:
For the first problem: You are given the speed and distance from the earth at a specific moment in the life of this spacecraft. The speed allows you to calculate the KE; and the distance allows you to calculate the PE of the system. It's that simple. (The stuff about 10 days is just to add a little background to the problem; conservation of energy will not help you answer the questions.)

For the second problem: Yes, you probably made an algebraic error; show what you did and we can take a look. (Hint: Find the speed at $r = \alpha R_e$ and compare it to the speed at $r = R_e$. Divide one expression by the other to get the ratio.)
So here are the speeds at both:
$$s_{e} = \sqrt{\displaystyle{\frac{2M_{e}G}{R_{e}}}}$$
$$s_{\alpha}=\sqrt{\displaystyle{\frac{2M_{e}G}{R_{e}\alpha}}}$$

Now, I have to express $$s_{\alpha}$$ in terms of $$s_{e}$$ and $$\alpha$$

...You mentioned that I should take a ratio of the two? But when I do that, I get:
$$\displaystyle{\frac{s_{e}}{\sqrt{\displaystyle{\frac{2M_{e}G}{R_{e}\alpha}}}}}$$

I know that this is probably just my messed up algebra.

EDIT: I figured out my error with the km/h to m/s conversion-- everything is fine with that one. THANKS! I also figured out the ratio problem!! THANKS!

Thank you sir!

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