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Part A

If the car was initially moving at a rate of 2.5 m/s and the rain is falling straight down, what is the speed of the car 2.0 minutes after the rain has begun?

Solution

As remarked above, because the rain has only a y-velocity initially, we can treat the process of collecting the rain as a typical collision in the x-direction, with momentum conserved. Thus, we can effectively diagram the situation as shown below.

Since there is no friction and the track is level, any change in the speed of the car must be due to the rain. By including the rain in our system, we have ensured that all x-forces are internal. Thus, the x-momentum must be a constant. We can therefore write:

\begin{large}\[ p^{\rm system}_{x,i} = p^{\rm rain}_{x,i} + p^{\rm car}_{x,i} = p^{\rm system}_{x,f} =  p^{\rm rain}_{x,f} + p^{\rm car}_{x,f} \] \end{large}

We define the initial time to be the instant before it begins to rain, because we have information about the car's velocity at that point. The final time is taken to be 2.0 seconds after the rain begins. Before the rain has accumulated in the car, it has zero x-velocity and hence zero x-momentum. After it has accumulated in the car, it is moving with the same speed as the train car. Thus, we can write:

\begin{large} \[ m^{\rm car}v^{\rm car}_{x,i}  = \left(m^{\rm car} + m^{\rm rain}\right)v_{x,f} \] \end{large}

All that remains is to determine the mass of the rain. We can do this by noting that the density of water is 1000 kg/m³ and that the water has filled the car to 2.0 cm deep, indicating a collected mass of:

\begin{large}\[ m^{\rm rain} = \rho^{\rm water}V^{\rm rain} = (\mbox{1000 kg/m}^{3})(\mbox{10 m}\times\mbox{3 m}\times\mbox{0.02 m}) = \mbox{600 kg} \]\end{large}

where ρ^water is the density of water and V^rain is the volume of the accumulated rain.

We can now solve to find:

\begin{large} \[ v_{x,f} = \frac{m^{\rm car}v^{\rm car}_{x,i}}{m^{\rm car} + m^{\rm rain}} = \mbox{1.6 m/s}\]\end{large}

Part B

Consider again the situation of Part A. Suppose that the rain is falling with a speed of 10.0 m/s, and that it comes to rest with respect to the train car immediately upon impact. At the instant the rain begins to hit the car, what is the force exerted on the train car by the rain?

Solution