A railroad car of length L and mass $$m0$$ when empty is moving freely on a smooth horizontal track while being loaded with sand from a stationary chute at a rate $$dmdt=q$$ Knowing that the car was approaching the chute at a speed $$v0$$. $$(Initially$$ at t $$=$$ $$0sec)Total$$ length of the sand equal to the Railroad car length L. DetermineThe speed of the car vf at the instant when the car has cleared the chute is $$V0e$$−$$2qLnm0v0$$. The value of n is

Question

A railroad car of length L and mass $$m0$$ when empty is moving freely on a smooth horizontal track while being loaded with sand from a stationary chute at a rate $$dmdt=q$$ Knowing that the car was approaching the chute at a speed $$v0$$. $$(Initially$$ at t $$=$$ $$0sec)Total$$ length of the sand equal to the Railroad car length L. DetermineThe speed of the car vf at the instant when the car has cleared the chute is $$V0e$$−$$2qLnm0v0$$. The value of n is

Infinity Learn · Accepted Answer

Let v be the velocity of the car when it has cleared x–distance as shown in figure andlet m be the mass of it (including$$ the $$load) at this instant. Then thrust force on it (dueto$$ change in $$mass) will be qv in backward direction. HenceFt $$=$$ qv or ma $$=$$ –qvor $$mvdvdx=$$−qvFrom conservation of linear momentummv $$=$$ $$m0v0Hence$$  $$m0v0dvdx=$$−qvSolving this we get  $$Vf=V0e$$−$$qLm0v0$$

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