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Physics, Classical Mechanics
Other
Orbital Mechanics
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An astronaught drifts away from an orbiter while repairing an antenna. The astronaught realizes the orbiter is moving away from him at 3m/s. The astronaught and his maneuvering unit have a mass of 100 kg.which includes a pressurized tank of of mass 10 kg. The tank contains only 2 kg of gas, in addition to the mass of the tank, that is used to propel him in space; the gas escapes at a constant velocity of 100m/s.

1. Will the astronaught run out of gas before he reaches the orbiter( at what point will he run out of gas)

2. With what velocity will the astronaught have to throw the empty tank away to get back to the orbiter?

Answer:

1.

The space is to be considered gravity free. As the orbiter is drifting away with slow speed of 3 m/s, the astronaught will be leaving behind the orbiter. To get back to the orbiter the astronaught must have a speed slightly more then 3 m/s towards the orbiter and for this the gas in the pressurized tank is used. As the gas is pushed back it imparts momentum to the astronaught and it moves towards the orbiter.

The total initial mass of the astronaught and the kit m0 = 100 + 10 + 2 = 112 kg

Let the total mass at some time t will be m and the velocity of astronaught be v then the momentum of the astronaught with the kit = m*v

Let at this instant in infinitesimally small time dt the mass escapes from the tank is dm, then the mass at time t + dt will be (m - dm)

Let the velocity of the astronaught increases by dv in this time interval dt then the velocity of astronaught at time t + dt will be (v + dv)

Momentum of the astronaught at time t + dt will be = (m - dm)*(v + dv)

Now velocity of the escape of the gas relative to the tank (backward) is u = - 100 m/s

In time dt the mass dm of the gas escapes from the tank with velocity u relative to the astronaught.

We know that the velocity of object a relative to b is given by

Hence if the actual velocity of the gas after escape is ve then we have the relative velocity u as

Gives
Whose magnitude is ve = -100 + v

Hence momentum of the gas escaped in time dt will be = dm*(-100 + v)

Now as there is no external force on the system, according to law of conservation of momentum the change in momentum of the system (total mass m) must be zero and hence we have

Final momentum - initial momentum = 0

Or (m - dm)*(v + dv) + dm*(- u + v) - m*v = 0

or m*v + m*dv - v*dm - dm*dv - u*dm + v*dm - m*v = 0

or m*dv - u*dm = 0 {neglecting dm*dv as it is comparatively very small}

or

This equation will give the increase in velocity of the astronaught in time dt. Integrating this equation with proper limits we will get the velocity of the astronaught when all the gas escape as

Or

Or m/s

Hence when the gas runs out the astronaught will have a velocity of 1.802 m/s, still less then the velocity of the orbiter and will not be able to reach the orbiter.

2.

Now to have a velocity just equal to that of the orbiter let he throws the tank with velocity vT, then again applying law of conservation of linear momentum we have

Initial momentum = final momentum

Or 110*1.802 = 100*3.0 + 10*vT

Or vT = - 10.178 m/s

Hence to have a velocity grater then the orbiter the astronaught must throw the empty tank with velocity grater then 10.178 m/s in the direction opposite to the orbiter.