Conservation of momentum changing mass

Posted on 2006-04-13
Last Modified: 2012-08-13
A railroad car filled with sand has a mass 85,000kg and is moving at 15m/s. After 20min 13,000kg has leaked out what is the speed?

m1 = 85,000kg
m2 = 72,000kg

v1 = 15m/s
v2 = ?

m_1 * v_1 = m_2 * v_2

v_2 = (m_1 * v_1)/m_2

v_2 = 17.70 m/s

Is this correct, if not why?
Question by:BrianGEFF719
    LVL 84

    Accepted Solution

    I'm not sure what happens to the sand when it leaks out, but if it is moving with the same velocity as the car when it leaves the car, and then has no further interactions with the car,
    I would think the velocity of 15m/s would not change.

    Or maybe the sand is suposed to hit the ground and stop moving, (losing momentum as momentum is transfered to the ground)
    and maybe you're then supposed to find the velocity of the centre of mass of the car+leaked sand?
    LVL 19

    Author Comment

    I think they want you to assume the falling sand is decreasing the mass of the system and the sand is no longer involved in the system.

    LVL 84

    Expert Comment

    Conservation of momentum only applies to a whole isolated system.
    Momentum in parts of a system can change by being transferred to other parts of a system.

    I still see no reason why sand leaking out of the car should affect the speed of what's left behind in this situation.

    If the sand that leaks out keeps its velocity after leaving the car, then
    85,000kg * 15m/s = 72,000kg * v2 + 13,000kg * 15m/s

    If 13,000kg of sand lose their momentum after leaving the car
    72,000kg * v2 = 85,000kg * 15m/s - 13,000kg * 15m/s

    If the sand is shot backwards out of the car with enough force so that its veocity relative to the ground is 0 when it leaves the car, then
    85,000kg * 15m/s = 72,000kg * v2 + 13,000kg * 0m/s
    LVL 19

    Author Comment

    So we cant say the car is an isolated system and the mass of that system is changing?

    LVL 4

    Expert Comment

    If the system is frictionless, nothing will change.  Suppose the railroad car was drifting through space.  You could shove out the sand, and it woud just coast along with the car, nothing would change.  If you want to change the speed of the car, you need to eject the sand forcibly out like a rocket.

    However, there is another way to look at this problem.  Perhaps we should assume that the engine is continuously pressing forward to overcome friction, with a constant force.  IF the main friction is wind resistance and does not change with the weight of the car, the speed will still remain unchanged when some sand leaks out.  But if the main friction is from the weight of the car on the track, the speed will increase.  However, to say how much you must model the total friction and how it changes with weight and speed.
    LVL 84

    Expert Comment

    The mass of an isolated system cannot change.
    (well, rest mass can change to energy, but the energy carries just as much momentum as the mass, so momentum still does not change)

    Expert Comment

    was working on this long enough, so thought i'd share

    m = initial mass
    dm = change in mass (is negative)
    dv = change in velocity
    v = instant velocity of wagon

    mv = (m + dm)(v + dv) + (-dm)(v + dv)
    mv = mv +mdv + vdm + dmdv - vdm - dmdv
    mdv = 0

    but m = 85000 kg, so dv = 0 !! velocity remains constant !!


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