A sand dropper drops sand of mass m(t) on a c | vedclass

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(D) The power $P$ required to maintain a constant speed $v$ for a conveyor belt onto which mass is being dropped is given by $P = F \cdot v$.Since the

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$P^2 \propto v^5$

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(D) The power $P$ required to maintain a constant speed $v$ for a conveyor belt onto which mass is being dropped is given by $P = F \cdot v$.Since the belt moves at a constant speed,the force $F$ required to overcome the momentum change of the added sand is $F = \frac{dp}{dt} = v \frac{dm}{dt}$.Given that the rate of mass deposition is $\frac{dm}{dt} = k\sqrt{v}$,where $k$ is a constant.Substituting this into the force equation: $F = v(k\sqrt{v}) = kv^{3/2}$.Now,calculating the power: $P = F \cdot v = (kv^{3/2}) \cdot v = kv^{5/2}$.Squaring both sides of the equation,we get $P^2 = k^2 v^5$.Therefore,$P^2 \propto v^5$.

$A$ sand dropper drops sand of mass $m(t)$ on a conveyer belt at a rate proportional to the square root of speed $v$ of the belt,i.e.,$\frac{dm}{dt} \propto \sqrt{v}$. If $P$ is the power delivered to run the belt at constant speed,then which of the following relationships is true?

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