A harvesting machine operates at a constant forward speed $v$ through a field. The rate at which it

Question

A harvesting machine operates at a constant forward speed $v$ through a field. The rate at which it gathers crop mass, $\frac{\mathrm{dm}}{\mathrm{dt}}$, is proportional to the square of its speed. If $P$ is the power supplied to the machine to sustain this constant speed, determine the correct proportionality between $P$ and $v$.

Professor PC Thomas Classes · Answer

Let $\boldsymbol{m}$ be the mass of the crop gathered by the machine. The machine operates at a constant forward speed $\boldsymbol{v}$. The rate at which it gathers crop mass is given as $\frac{\mathrm{dm}}{\mathrm{dt}} \propto \boldsymbol{v}^2$. We can write this as $\frac{\mathrm{dm}}{\mathrm{dt}}=k \boldsymbol{v}^2$, where $\boldsymbol{k}$ is a constant of proportionality.
To sustain a constant speed $\boldsymbol{v}$, the machine must exert a force to accelerate the incoming crop mass from rest to speed $\boldsymbol{v}$. This is a classic variable mass system problem. The force $\boldsymbol{F}$ required is given by Newton's second law for a variable mass system, considering the relative velocity of the incoming mass. Since the crop is initially at rest relative to the ground and is accelerated to the machine's speed $\boldsymbol{v}_{\boldsymbol{r}}$ the force exerted by the machine on the crop (or the reaction force on the machine) is:
$$
F=v \frac{\mathrm{dm}}{\mathrm{dt}}
$$Substitute the given proportionality for $\frac{\mathrm{dm}}{\mathrm{dt}}$ :
$$
\begin{aligned}
& F=v\left(k v^2ight) \
& F=k v^3
\end{aligned}
$$The power $\boldsymbol{P}$ supplied to the machine to sustain this constant speed is given by the product of the force and the speed:
$$
P=F \cdot v
$$Substitute the expression for $\boldsymbol{F}$ :
$$
\begin{aligned}
& P=\left(k v^3ight) v \
& P=k v^4
\end{aligned}
$$Therefore, the power $\boldsymbol{P}$ is proportional to $\boldsymbol{v}^{\mathbf{4}}$.
$$
P \propto v^4
$$The correct option is C.