How Much Velocity Can a Spacecraft Gain by Shooting Xenon Ions?

[sander2798]

Hello everyone,

the problem I have to solve is as following.

You're in space and you're moving with a constant speed. Now, you want to accelerate. For this you will be using Newton's third law.

Xenon 1+ iones will be shot from the back of the spacecraft with high speeds, the avarage speed an ion will be accelerated to is 30.000 m/s. This will result in a gain in velocity of the spacecraft , according to Newton's 3rd law. Calculate how many velocity a spacecraft with a mass of 1000 kg can gain by shooting away 50kg of Xenon 1+ iones.

My attempt:

Mass of one Xenon +1 ion:
= 131.30 x 1.660538921 x 10^-27 = 2.18 x 10^-25 kg

Kinetic energy gain of ion:
1/2 mv^2 = 1/2* 2.18*10^-25 *30000^2 = 9.81*10^-17 Joule

According to Newton's third law the gain in kinetic energy of the ion should be equal to the gain in kinetic energy of the spacecraft , right? (not sure about this)

1/2 mv^2 = 1/2 * 1000 * v^2 = 9.81*10^-17
v = 4.43 * 10^-10 m/s for every ion shot

Calculating how many iones we have:
100000 / 131.30u = 761.614 mol.
761.614 * 6,022 * 10^23 = 4.586 * 10^26 iones.

Shooting away all those iones will create a gain in speed of:
4.586 * 10^26 * 4.43 * 10^-10 = 2.03 * 10^17 m/s

But since this is about a few thousand times the speed of light, I assume I did something wrong somewhere. If anyone could help me I'd be very thankful.

Sander.

 

[paisiello2]

According to Newton's third law the gain in kinetic energy of the ion should be equal to the gain in kinetic energy of the spacecraft , right? (not sure about this)

This is wrong. You should look up the definition on Wikipedia or something.

What you actually did here is apply the principle of conversation of energy. However. conservation of energy doesn't work the way you put it because you neglected the work of the magnetic field that accelerated the ions to begin with.

Can you think of something else that is conserved here instead?

Good for you, though, that you did a reality check at the end and realized you did something wrong.

 

[sander2798]

This is wrong. You should look up the definition on Wikipedia or something.

What you actually did here is apply the principle of conversation of energy. However. conservation of energy doesn't work the way you put it because you neglected the work of the magnetic field that accelerated the ions to begin with.

Can you think of something else that is conserved here instead?

Good for you, though, that you did a reality check at the end and realized you did something wrong.

Thanks, now have the correct answer! (I hope :P)