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

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 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?

• The textbooks is correct; your reasoning is wrong because the sand that is falling out is clearly carrying away KE and so KE must be decreasing. Although your problem is conceptual, this homework problem is specifically testing your conceptual understanding, so this is actually against site rules. Commented Aug 8 at 4:15
• @naturallyInconsistent But I did my own work, didn't I? How is that against the rules? Reported your comment for being 'rude'. Commented Aug 8 at 4:19
• This post is focused on a conceptual question framed around a homework problem. The OP isn’t asking for someone to outline a solution, nor to check work. If this isn’t an example of using an example to ask a conceptual question, then I am not sure what is. I am voting to reopen. Commented Aug 8 at 10:35
• @BioPhysicist The last line containing the question is mostly a "where is my mistake" question to me. Commented Aug 8 at 12:32
• @VincentThacker You could tack that onto any question... "I don't understand why objects have the same acceleration due to gravity. If something is heavier, isn't it pulled more? Where is my mistake? Is my reasoning correct?" The OP has outlined a conceptual misunderstanding here. The closure reason isn't "where is my mistake" question Commented Aug 8 at 13:39

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.