Question

For power to be constant, the force has to vary with speed as .

Answer 1

Hint: Write the formula of instantaneous power consumed. In that, replace the formula of work and differentiate. We will get a formula relating the three physical quantities i.e. power, force, and velocity. We can then proceed from there.

Complete step by step answer:
The power is defined as the rate at which the work is done or energy is consumed. If a force does an amount of work $W$ in an amount of time $\Delta t$, the average power due to the force during that interval is ${P_{avg}} = \dfrac{W}{{\Delta t}}$ .
The instantaneous power is defined as the instantaneous time rate of doing work. Mathematically, this can be written as- $P = \dfrac{{dW}}{{dt}}$.
For a particle that is moving along a straight line in the X-axis and acted by a constant force. The value of work done $W$ is given by : $W = Fx\cos \theta $.
If we substitute this value of work done in instantaneous power formula ,we get :
$\implies P = \dfrac{{d(Fx\cos \theta )}}{{dt}}$
$\implies P = \dfrac{{d(Fx\cos \theta )}}{{dt}}$
$\implies P = F\cos \theta \dfrac{{dx}}{{dt}}$
Since, $v = \dfrac{{dx}}{{dt}}$
$\implies P = F\cos \theta v$………..(i)
where $\theta $= angle between Force and Velocity.
Power, force and velocity vary with each other with the relation shown in the above equation.
Now, if we keep the power of the body constant. Let's say power $P = k$ (here, constant). Then from equation (i), we have
$P = Fv\cos \theta = k$
$
\implies Fv\cos \theta = k \\
\implies F = \dfrac{k}{{v\cos \theta }} \\
$
$\implies F \propto \dfrac{1}{v}$……….(ii)
From equation (ii), it is clear that if power is kept uniform, the force and velocity vary inversely with each other.

Note: Power is a scalar quantity i.e. it has only magnitude and no direction. The power of an electrical appliance tells us the rate at which electrical energy is consumed. The S.I. unit of power is Watt. Another famous unit of power is ‘horsepower’ .This unit originated back when steam engines first replaced ‘horses’ as a source of power.