Suppose you are navigating a spacecraft far from other objects. The mass of the

Question

Suppose you are navigating a spacecraft far from other objects. The mass of the spacecraft is 4.0 104 kg (about 40 tons). The rocket engines are shut off, and you're coasting along with a constant velocity of ‹ 0, 28, 0 › km/s. As you pass the location ‹ 7, 6, 0 › km you briefly fire side thruster rockets, so that your spacecraft experiences a net force of ‹ 4.0 105, 0, 0 › N for 21 s. The ejected gases have a mass that is small compared to the mass of the spacecraft. You then continue coasting with the rocket engines turned off. Where are you an hour later? (Think about what approximations or simplifying assumptions you made in your analysis. Also think about the choice of system: what are the surroundings that exert external forces on your system?)

Explanation / Answer

Let's think about we know to begin with:

Our spaceship has a mass of 4.0 * 10^4 kg, 40 metric tons.

Our initial velocity is a vector, v0 = <0,28,0> km/s

At t=0 we pass a point x0 = <7,6,0> km

The net force we place on our rocket by the thrusters is also a vector:

F = <4.0* 10^5 N, 0, 0> for 21 s

This impuse can also be written as a vector:

J = <8.4 * 10^6 Ns, 0, 0>

We need to calculate our change in momentum that the rocket experiences to find our final velocity.

Initial momentum:

P0 = <0, 28000 m/s * 40000 kg, 0>

P0 = <0, 1.12 * 10^9 Ns, 0>

J = <8.4 * 10^6 Ns, 0, 0>

Since P1 = P0 + J,

P1 = <8.4 * 10^6, 1.12 * 10^9, 0> Ns

From our momentum we can find our final velocity, since P=mv.

v1 = <210, 28000, 0> m/s

Since x0 = <7000, 6000, 0> m, and x1 = x0 + v*t we can find our final position. After an hour (3600 s) we will be in the final position x1:

x1 = <7000 + 756000, 6000 + 100800000, 0> m

x1 = <763000, 100806000, 0> m

x1 = <763, 100806, 0> km

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