Irodov Problem 1.180

This problem is a straightforward variation of 1.179 where v=wt and we have to determine at what rate the mass changes. One thing to note here is that the escape velocity of the gas u in this problem is not bold and is a scaler. This may be a bit confusing especially since in 1.178 and 1.179, he uses the bold vector notation. The essential idea in the notation is simply that when he uses vector notation, if the direction of ejection is in the opposite direction to that of the rocket, the corresponding scalar will only be negative in other words,




So now using the scalar u instead of the vector notation from 1.179 we have,

Irodov Problem 1.179

We can re-write the expression derived in 1.178 as,






So integrating both sides we have,

Irodov Problem 1.178

The rate of change of momentum of the (rocket+fuel) system is equal to the sum of all the forces acting on the (rocket+ fuel) system.

Suppose that at some instant of time the mass of the rocket was . Suppose that over the next infinitesimally small interval of time dt, fuel of mass shoots out if the rocket. The new mass of the rocket is thus now m-. Suppose the during this time the velocity of the rocket increases by dv, so its velocity is not v+dv. The velocity of the fuel relative to the rocket is u (this is a vector in the upward direction) relative to the rocket, so the velocity of the fuel relative to a stationary reference frame is v+u. The net change of momentum of the (rocket +fuel) system will be Fdt.

The change in momentum dP of the entire (rocket + fuel) system is given by,

Irodov Problem 1.177

As with the last few problems, momentum will be conserved before and after collision of the molecules and so we have,























In case the collision is elastic no energy will be lost and so we have,









Here is the angle of divergence. One trivial solution to (6) is . This corresponds to a perfectly head on collision, in this case the angle of divergence will be 0. Now let us consider the non-trivial case when .









Now (7) can only be true if . This demonstrates part a) of the question.

In case of an inelastic collision we will have,





Clearly then and this demonstrates part b).