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# "Kinetic Energy and Collisions"

A new communications satellite is being sent into space. The rocket carrying the satellite has a mass of 2.04 x 10^6 kg. The engines expel 3.7 x 10^3 kg of exhaust gas during the first second of liftoff giving the rocket a velocity of 5.7 m/s [up]. At what velocity is the exhaust gas leaving the rocket engines? Ignore the change is mass due to the fuel being consumed. The exhaust gas needed to counteract the force of gravity has already been accounted for and should not be part of this calculation. Show all calculations.

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Linear momentum conservation Law is required to answer this question.

In the absense of external forces, the linear momentum of a system is conserved.

Linear momentum of a body with mass m is mV, were V is the velocity of the body.

The linear momentum of rocket-gas system prior to expulsion is 0 because it is at rest.

The linear momentum of rocket-gas system after the expulsion is (M-m)V - mU. Here M is the mass of the rocket, m is the mass of the gas, V is the velocity of the rocket (5.7 m/s) and U is the unknown velocity of the gas, which is opposite in direction to V.

Since mass of gas m is thousand times smaller than mass of rocket M, it can be neglected in expression M-m. So we can write the momentum of the system after the expulsion as

MV - mU = 0.

From here U = MV/m = 2.04 * 10 ^6 * 5.7 / (3.7 * 10^3) m/s =

= 3.14* 10 ^3 m/s

The exhaust gas leaves the rocket with the velocity of 3.14*10^3 m/s.

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