Animated Solution for Physics - Work, Energy, and Power: A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time $t$ is proportional to

The concept of constant power is a classic trap for many students. When we hear "constant power," our brains often jump to "constant force" or "constant acceleration." But as we'll soon discover, constant power means the force must actually decrease as the object speeds up! Let's dive into the mathematics and physics behind this fascinating phenomenon.

Imagine you are pushing a heavy box across a frictionless floor. If you apply a constant force, the box accelerates at a constant rate. However, the power you are delivering is $P=Fv$. As the box gets faster (increasing $v$), you have to deliver more and more power to maintain that constant force.

But what if your machine has a maximum power output? What if it can only deliver a constant power $P$? In that case, as the velocity $v$ increases, the force $F$ must decrease to keep the product $Fv$ constant. This means the acceleration is not constant; it's decreasing over time!

Our Starting Point

We start with the fundamental definition of mechanical power for linear motion:

We know from Newton's Second Law that force is the product of mass and acceleration, $F=ma$. Furthermore, acceleration is the rate of change of velocity, $a=\frac{dv}{dt}$. Substituting this into our power equation gives us a beautiful differential equation:

The Work-Energy Shortcut

To find out how velocity changes with time, we need to solve this differential equation. We do this by separating the variables—putting all the $v$ terms on one side and the $t$ terms on the other:

Now, we integrate both sides. Assuming the body starts from rest ($v=0$ at $t=0$):

Solving for $v$, we get:

Pro Tip: There is an incredibly elegant shortcut to reach this exact same equation using the Work-Energy Theorem! The total work done by the machine in time $t$ is simply $W=Pt$. Since the body starts from rest, this work goes entirely into its kinetic energy: Physics is beautiful when different paths lead to the exact same truth!

From Velocity to Displacement

We have the velocity as a function of time, but the question asks for the distance moved. We know that velocity is the rate of change of displacement, $v=\frac{ds}{dt}$. Let's substitute this into our velocity equation:

Once again, we separate the variables and prepare for integration:

Integrating both sides from $t=0$ to $t$:

The integral of $t^{1/2}$ is $\frac{t^{3/2}}{3/2}$. Plugging this in, we get our final expression for displacement:

Look closely at our final equation. The terms $\frac{2}{3}$, $P$, and $m$ are all constants. Therefore, we can strip away the constants to reveal the core proportionality:

This result is profound. If the acceleration were constant, the distance would be proportional to $t^2$. If the velocity were constant, the distance would be proportional to $t$. Constant power sits right in the middle, giving us a $t^{3/2}$ dependence. Understanding this derivation not only secures you marks in JEE but also deepens your intuition for how machines actually operate in the real world!