Q 11-04-069JEE MainJEE Main 2025 (29 Jan, Shift 2)Medium
A sand dropper drops sand of mass $m(t)$ on a conveyor belt at a rate proportional to the square root of the speed $(v)$ of the belt, i.e. $\dfrac{dm}{dt} \propto \sqrt v$. If $P$ is the power delivered to run the belt at constant speed, then which of the following relationships is true?
Answer: (C) $P^2 \propto v^5$
To keep the belt at constant speed, the force must supply momentum to the falling sand:
$$F = v\frac{dm}{dt} \propto v\sqrt v = v^{3/2}$$
$$P = Fv \propto v^{5/2} \Rightarrow P^2 \propto v^5$$
Solution by Sreeraj P, M.Sc Physics