Sand Sprayer and conservation of linear momentum

The solution to a problem from "Introduction to Mechanics" by Kleppner and Kolenkow has created a doubt in my head.

A sand-spraying locomotive sprays sand horizontally into a freight car as shown in the sketch below. The locomotive and freight car are not attached. The engineer in the locomotive maintains his speed so that the distance to the freight car is constant. The sand is transferred at a rate $b=10 kg/s$ with a velocity $u=5 m/s$ relative to the locomotive. The freight car starts from rest with an initial mass of $M_0= 2000 kgs$. Find its speed after $t=100 s$.

The solution to the problem 4.13 is on page 61. I am in a belief that the solution is not correct.

The solution considers the momentum of the system( defined in the solution ) to be conserved. The problem I find with the solution is, the sand( part of the system ) is being accelerated by the locomotive( as the locomotive keeps up with the speeding freight car ), suggesting the presence of an external force( exerted by the locomotive ) on the system. The solution clearly states that the external force on the system is zero. What am I not understanding here?

• @Farcher the system consists of the freight car and some amount of sand. The sand being a part of the system is being ejected by the locomotive( external ). Shouldn't this amount of sand experience an external force? – R004 Jan 18 '18 at 7:32
• I had a lot to write so have written an answer. – Farcher Jan 18 '18 at 7:41
• What is your solution to the problem? Is it only the explanation which you disagree with, or do you get a different result? – sammy gerbil Jan 22 '18 at 16:38
• @sammygerbil the only part of the solution that bothers me is that the net external force on the system is considered to be zero. Here, the mass of the freight and some of the sand that eventually flows into the freight is taken to be the system. As the locomotive keeps up with the freight, I think that the sand being released is being given a push by the locomotive( as sand dunes speed up relative to the ground ). Don't you think that these dunes( part of the system ) are being acted upon by an external force, a force exerted by the locomotive( external )? – R004 Jan 22 '18 at 16:47
• Farcher has already tried to explain. I am not sure I could do any better. Sometimes the more you try to understand something the harder it is to understand. Then it is best to try a different approach. Try to provide your own solution. If that gives the correct answer, you have nothing to worry about. – sammy gerbil Jan 22 '18 at 17:14

There is no physical contact between the locomotive and the freight car.

Imagine that you were on the locomotive and threw a ball at the freight car.
Once the ball has left you hand you exert no further force on the ball.
Hence the momentum of the ball plus momentum of the freight car does not change as there are no external forces on the freight car and the ball.

The freight car speeds up and you speed up by the same amount.
Your momentum and that of the locomotive has to change by having a force exerted on the locomotive and you but that force is not exerted on the freight car.
You throw another ball at the freight car at the same speed relative to the locomotive (and freight car) and again once the ball has left your hand you exert no further force on the ball.
Once again the ball plus the freight car can be considered as a system with no external forces acting on it with their change in momentum zero.

The sand can be treated as a continuum of balls that you are throwing at the freight car.

Your concern was related as to how the sand got to a certain speed relative to the locomotive and hence the freight car.
That is irrelevant in this problem as once the sand has achieved that speed no further forces need to act on the sand and those forces do not act on the freight car.

• Let me say that at $t=0$, the car is at rest and I, from the locomotive, throw two balls in a succession. The car plus two balls constitute my system. Once the first ball leaves my hand with a velocity $u$ relative to me( and the ground ), I start the timer. Say that the second is at rest w.r.t to me. The first ball lands and the car begins to move with $v$( say ). I match its speed. Now, I throw the second. Here I have a problem. The second is accelerating from $0$ to $v+u$ within the time of interest. Does this not suggest that an external force is acting on the system? – R004 Jan 18 '18 at 11:43
• @R004 No. There is no way that the freight car “knows” what is/has happening to the second ball before the second ball hits the freight car. – Farcher Jan 18 '18 at 13:08
• that is true. But the second ball is a part of my system. There is a force on the second ball, is there not? – R004 Jan 18 '18 at 13:23