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

Constant Power in Action

Imagine you are standing on a perfectly smooth horizontal surface, and right in front of you is a block resting quietly. Its initial velocity is zero. Suddenly, a machine engages and starts pushing this block. But here is the catch—it doesn't push with a constant force; it pushes with a constant power.

What does that mean physically? Power is the rate at which work is done, or the product of force and velocity ($P=Fv$). If the power is constant, as the block speeds up, the machine must push with less and less force to maintain that constant energy flow. This is a classic scenario in physics that requires us to bridge the gap between energy and kinematics.

The First Bridge

To find out how far the block travels, we first need to know how fast it is going. This is where the Work-Energy Theorem comes to our rescue. It states that the net work done on an object equals its change in kinetic energy.

Since the machine delivers a constant power $P$ over a time $t$, the total work done is simply the product of power and time:

Now, we equate this work done to the change in kinetic energy. Since the block started from rest, its initial kinetic energy is zero. Therefore, the final kinetic energy is entirely due to the work done by the machine:

From Velocity to Displacement

With our energy equation set up, we can easily isolate the velocity $v$. Rearranging the terms, we get:

Taking the square root of both sides gives us velocity as a function of time:

This is a beautiful result! It tells us that the velocity grows with the square root of time. But we are not done yet. The question asks for the displacement $s$. We know from basic kinematics that velocity is the rate of change of displacement:

Substituting this into our velocity equation, we set the stage for the final mathematical operation:

To find the total distance moved, we need to sum up all the tiny displacements $ds$ over the time interval from $0$ to $t$. This means we must integrate both sides of our equation:

Integrating $t^{1/2}$ is a straightforward application of the power rule. We add $1$ to the exponent and divide by the new exponent:

Simplifying the constants, we arrive at our master equation for displacement:

Since $\frac{2}{3}\sqrt{\frac{2P}{m}}$ is entirely composed of constants, we can clearly see the proportional relationship:

And there we have it! The distance moved by the body is proportional to $t^{3/2}$. It is fascinating to contrast this with a constant force scenario, where acceleration is constant and $s\propto t^2$. The constant power constraint fundamentally alters the kinematic trajectory, making it a favorite concept for competitive exams!