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

Imagine you are standing on a smooth, frictionless horizontal surface. A box is resting quietly in front of you. Suddenly, a machine hooks onto it and starts towing it along a straight line.

The defining characteristic of this machine is that it delivers constant power.

This is a crucial detail. It doesn't deliver constant force, nor does it maintain a constant velocity. The rate at which it pumps energy into the system is perfectly steady. Our mission is to figure out how the distance $x$ moved by this box depends on the time $t$.

To crack this, we need to translate the physical situation into mathematical language. What exactly is power in mechanics?

Power $P$ is defined as the rate at which work is done. When a force $F$ moves an object at a velocity $v$, the power delivered is the dot product of the two. Since the motion is strictly along a straight line, we can write this simply as:

We are given that $P$ is a constant. Now, according to Newton's Second Law, the net force acting on an object is equal to its mass $m$ multiplied by its acceleration $a$.

Let's substitute this into our power equation:

Here is where we need to be careful. Because the product of acceleration and velocity is constant, as the box speeds up (velocity increases), its acceleration must decrease!

This means we absolutely cannot use the standard kinematic equations like $s=ut+\frac{1}{2}a{t}^2$. Those are strictly reserved for constant acceleration. Instead, we must rely on the fundamental definitions of motion using calculus.

We know that acceleration is the rate of change of velocity:

Substituting this back into our power equation gives us a differential equation:

To solve this, we separate the variables. We keep all the $v$ terms on one side and move the $t$ terms to the other:

Now, we integrate both sides. The box was initially at rest, so at $t=0$, $v=0$.

Solving for $v$, we get:

Since $P$ and $m$ are constants, we can clearly see that the velocity is proportional to the square root of time:

We are almost there! We have the velocity as a function of time, but the question asks for the distance $x$.

Velocity is simply the rate of change of position:

So, we can write:

Once again, we separate the variables and integrate to find the position $x$:

Using the power rule for integration, we add $1$ to the exponent and divide by the new exponent:

Since $3/2$ is just a constant number, it gets absorbed into the proportionality. We arrive at our final, elegant result:

The distance moved by the box is directly proportional to $t^{3/2}$. This perfectly matches option (b).

This is a classic and highly important result in physics. Whenever a system is driven by a constant power source, the kinetic energy grows linearly with time, the velocity grows with the square root of time, and the distance grows with the three-halves power of time!