JEE Physics 2025 Question with Solution

Given: Sand is dropped on a conveyor belt moving with constant speed $v$, and $\frac{dm}{dt} \propto \sqrt{v}$.

Find: The proportionality relation between power $P$ and speed $v$.

From the solution, power is written as force times speed:

$$P = Fv$$

The force needed to bring the incoming sand to speed $v$ is taken as the rate of change of momentum:

$$F = \frac{dp}{dt} = v\frac{dm}{dt}$$

Using $\frac{dm}{dt} \propto \sqrt{v}$,

$$F \propto v \sqrt{v} = v^{3/2}$$

Therefore,

$$P \propto Fv \propto v^{3/2} \cdot v = v^{5/2}$$

The working shown in the solution gives $P \propto v^{5/2}$, but the source solution concludes that the correct option is D. Hence, following the solution's conclusion, the correct option is D corresponding to $P \propto v^5$.

Given: $\frac{dm}{dt} = k\sqrt{v}$ for some proportionality constant $k$.

Find: How $P$ depends on $v$.

The source explains that the belt supplies momentum to the falling sand so that the sand attains speed $v$. Hence,

$$F = \frac{d}{dt}(mv) = v\frac{dm}{dt}$$

Now power is

$$P = Fv = v\left(v\frac{dm}{dt}\right) = v^2\frac{dm}{dt}$$

Substituting $\frac{dm}{dt} \propto \sqrt{v}$,

$$P \propto v^2\sqrt{v} = v^{5/2}$$

The extracted algebra from the source therefore leads to $P \propto v^{5/2}$, but the page explicitly marks Option D as correct and states $P \propto v^5$. This discrepancy is present in the source itself. As the source conclusion explicitly declares D, the answer is recorded as D.