Do I use the law of inertia or the work-energy theorem?

Question

My textbook (Physics Part I: Textbook for Class XI, NCERT) has the following question:

A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of 0.05 kg/s. What is the speed of the trolley after the entire sand bag is empty?

The answer given here is:

The system of trolley and sandbag is moving with a uniform speed. Clearly, the system is not being acted upon by the external force. If the sand leaks out, even then no external force acts. So there shall no change in the speed of the trolley. (sic)

Fair enough, I get it. But according to the work-energy theorem, as no external force is acting on the trolley, the kinetic energy should remain constant. But the mass is changing from the initial state to the final state, and this means the speed should change so that the KE could indeed remain constant.

My textbook says that the speed shouldn't change.

Judging by my textbook's answers page and everywhere I look, it seems my reasoning is incorrect. How is that?

Answer 1

Your KE decreases because you are losing mass. Your linear momentum is conserved because there are no external forces acting on your trolley+sand system.

Consider your trolley initially has mass $M$, with speed $v$. After it started to leak sand, imagine the instant it lost sand mass $dm$, which was also moving at speed $v$ due to inertia. Conservation of linear momentum gives $$Mv = (M-dm)v'+vdm$$ Clearly, $v'=v$ and your speed is unchanging. Also, the kinetic energy of your trolley alone (not the lost sand) is $$KE = \frac{Mv^2}{2} \Rightarrow \frac{d(KE)}{dt} = \frac{v^2}{2}\left(\frac{dM}{dt}\right)$$ Since you are losing mass, $\frac{dM}{dt} < 0$, so $\frac{d(KE)}{dt}<0$ and you lose KE as well.

Answer 2

At the instant a grain of sand drops from the trolley, it still has the same forward velocity. Presumably, it falls and comes to rest. So there is an external force on each grain of sand that does not act on the trolley. This force reduces the sand KE to zero but not the trolley KE. As previously stated there are two ways to lose kinetic energy: by losing velocity or by losing mass.