"Variable mass" problems

Problem:

A trainload of empty, non-leaking coal cars is accelerating in an easterly direction due to a constant force F.  It starts to rain, and the rain has a velocity which has a horizontal component v0, in a westerly direction.  The rain fills the cars at a constant rate of λ mass units per unit time.  Under these conditions the train will reach a maximum velocity. 
(a)  Find the maximum velocity and express it in terms of F, v0, and λ.
(b)  Find the train's velocity v(t).

Solution:

Problem:

Water starts emerging horizontally with speed v0 from a nozzle and just reaches the far end of a box with mass m0 and length l which rests on a frictionless surface.
The mass flow rate of the water is λ.

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Find the velocity of the box as a function of time as long as water pours into the box.

Solution:

Problem:

A spherical water droplet falls without friction under the influence of gravity through an atmosphere saturated with water vapor.  Let its initial radius at t = 0 be C, its initial velocity v0.  As a result of condensation, the water drop experiences a continuous increase in mass, proportional to its surface.  Its radius R then increases linearly with time.  Integrate the differential equation of the motion by introducing R instead of t as independent variable.  Show that for C = 0 the velocity increases linearly with time.

Solution:

Problem:

A spacecraft of mass m0 and cross-sectional area A is coasting with velocity v0 when at t = 0 it encounters a stationary dust cloud of density ρ.  If the dust sticks to the spacecraft, solve for the speed of the spacecraft as a function of time.  Assume A is constant over time.

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Solution:

Problem:

The UT Physics Department's vacuum cannon fires a 3.8 cm diameter, 2.3 g ping pong ball through a 1.8 m long pipe that has been evacuated to a pressure which is a very small fraction of atmospheric pressure, so that its initial acceleration is simply determined by Newton's second law and the force produced by one atmosphere of pressure acting on the cross-sectional area of the ping pong ball.  (The ends of the pipe are sealed with a very thin plastic film.)  However, as the ball accelerates down the pipe, the atmospheric pressure-force also has to accelerate the mass of air that rushes in to fill the pipe behind the ball.  Assume that this variable mass piston of air is a cylinder having the same density as the surrounding atmosphere (1.3 kg/m3) and an initial mass of zero, and that none of this air leaks around the edges of the ball into the evacuated space ahead of the ball. 
(a)  Neglecting the effect of the variable mass piston of air, i.e. assuming the acceleration to be constant along the length of the pipe and equal to the initial acceleration, constant acceleration, find the (incorrectly predicted) exit velocity of the ball.
(b)  Derive an expression for the speed of the ball as a function of distance accelerated down the cannon, taking the variable mass piston of air into account. 
(c)  Evaluate the terminal velocity (for a very long cannon) for the numerical values given above, and the predicted velocity for our version of the cannon, which is 1.8 m long.  Compare these values, and the result of (a), with the speed of sound in air at room temperature (about 340 m/s) and discuss that comparison.

Solution: