When a constant force is applied to a body moving with a constant acceleration, is power of the force constant? If not, how will force vary with speed for the power to be constant?

Obtaining the relationship between power, force, and velocity.

Power is given as,

$P=\frac{W}{t}$

where, $W$ is work done and $t$ is time.

Work done is given as,

$W=Fd$

where $F$ is force and $d$ is displacement.

Substituting $Fd$ for $W$ in the equation for power, we get,

$P=\frac{Fd}{t}$

Now, speed is given as,

$v=\frac{d}{t}$

where $v$ is speed or velocity, $d$ is displacement and $t$ is time.

Therefore, the equation for power can be written as,

$$\begin{gathered} P=F\left(\frac{d}{t}\right) \\ P=Fv\left[\because v=\frac{d}{t}\right] \end{gathered}$$

Explanation

1. Acceleration can be defined as the rate of change in velocity. Since the body has constant acceleration, it can be said that there is an increase in the velocity of the body.
2. As velocity changes, power also changes. Therefore, the power of the force is not constant.
3. From $P=Fv$, it can be said that $F=\frac{P}{v}$.
4. If power is to be kept constant, it can be said that $F\propto \frac{1}{v}$

Therefore, if the body has acceleration, its velocity is changing, and in turn, power also changes. For the power to be constant, force should vary inversely to velocity. In other words, since velocity is increasing (due to acceleration), force should be decreased in order to keep the power constant.