Question

Let there be a tank-car filled with water, shown in figure-4.59. The initial mass of the car with water is M. At t = 0, a hole is made in the left wall of the car and water start spilling out from the car, with a constant velocity u with respect to the car. The rate of ejection of water is r kg/s. Find the velocity of the car as a function of time.

Answer 1

The velocity of the car as a function of time is: $v(t) = u \ln\left(\frac{M}{M-rt}\right)$

Answer 2

The problem is solved by applying the conservation of linear momentum to a system of variable mass. The rate of change of momentum of the system is equal to the momentum imparted by the ejected mass. The differential equation derived from this principle is integrated, assuming the car starts from rest, to obtain the velocity as a function of time.