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An isolated rail car originally moving with speed V o on a straight, frictionless, level track contains a large amount of sand. A release valve on the bottom of the car malfunctions, and sand begins to pour out straight down relative to the rail car. (a) Is momentum conserved in this process? (A) The momentum of the rail car alone is conserved. (B) The momentum of the rail car + sand remaining within the car is conserved. (C) The momentum of the rail car + all of the sand, both inside and outside the rail car, is conserved. (D) None of the three previous systems have momentum conservation. (b) What happens to the speed of the rail car as the sand pours out? (A) The car begins to roll faster. (B) The car maintains the same speed. (C) The car begins to slow down. (D) The problem cannot be solved since momentum is not conserved.

An isolated rail car originally moving with speed Vo on a straight, frictionless, level track contains a large amount of sand. A release valve on the bottom of the car malfunctions, and sand begins to pour out straight down relative to the rail car.
(a) Is momentum conserved in this process?
(A) The momentum of the rail car alone is conserved.
(B) The momentum of the rail car + sand remaining within the car is conserved.
(C) The momentum of the rail car + all of the sand, both
inside and outside the rail car, is conserved.
(D) None of the three previous systems have momentum conservation.
(b) What happens to the speed of the rail car as the sand pours out?
(A) The car begins to roll faster.
(B) The car maintains the same speed.
(C) The car begins to slow down.
(D) The problem cannot be solved since momentum is not conserved.

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
8 years ago
(a)
(A) The momentum of the rail car alone is conserved .
(b)
(A) The car begins to roll faster.
(a)
When no sand leaks out of the railcar, the net force experienced by the rail car + sand system in the vertical direction is zero, and therefore the rate of change of momentum of the system in the vertical direction is also zero.
Similarly, in the horizontal direction, the sand and the railcar move with a constant speed v0 initially, and no net force acts on them, and accounts for no change in momentum over time.
When the sand starts to leak out, the normal force from the railcar no longer exists, and the gravitational force pulls the sand downwards. The rate of change of momentum of sand in the vertical direction is equal to the gravitational force on the sand falling downwards.
Therefore, the rate of change of momentum in the vertical direction has changed from zero to the magnitude of gravitational force on the falling sand. From this, one can infer that the momentum of the rail car-sand system in the vertical direction is not conserved over time.
In spite the fact that the sand leaks out in vertical direction, no net force acts in the horizontal direction. Therefore the momentum of the rail car + sand system in the horizontal does not change as the sand falls down.
From above discussion, one can rule out options (B) and (C).
On the assumption that the system consist of the rail car alone, and the fact that no external force acts on the rail car initially, and even when the sand leak out, one can sat that the momentum of the railcar alone is conserved.
Therefore, (A) is the correct option, and the rest are ruled out.
(b)
Assume that amount of sand leaks out after time t while the total initial mass of rail car and sand is M.
Using this one can calculate the final horizontal speed say (v’) of the rail car + sand system as:

236-790_1.PNG
Therefore, the horizontal speed of the railcar and the remaining sand increases as the sand falls down.
Thus, (A) is the correct option, and the rest are ruled out.

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