Animated Solution for Physics - Work, Energy, and Power: A body of mass $m$ is accelerated uniformly from rest to a speed $v$ in a time $T$. The instantaneous power delivered to the body as a function of time, is given by

Unraveling Instantaneous Power

Imagine you are standing on a perfectly smooth, frictionless surface, and you start pushing a heavy block of mass $m$. You push it with a steady, unwavering effort, causing it to accelerate uniformly. It starts from a complete standstill, meaning its initial velocity $u$ is zero. After a certain time $T$, you've managed to get it moving at a brisk speed $v$.

The question we want to answer is: at any random moment $t$ during this journey, exactly how much power are you delivering to the block? This is what we call instantaneous power.

To find power, we first need to understand the force you are applying. Since the block is accelerating uniformly, we can easily calculate this acceleration using the fundamental kinematic equations. Acceleration is simply the rate of change of velocity.

Given that the block goes from $0$ to $v$ in time $T$, the uniform acceleration $a$ is:

Now, Newton's Second Law tells us that the force $F$ required to produce this acceleration is the product of mass and acceleration ($F=ma$). Therefore, the constant force you are applying is:

Power isn't just about force; it's about how fast that force is moving the object. We need the block's exact velocity at our specific time $t$. Let's call this instantaneous velocity $v(t)$.

Using the first equation of motion ($v=u+at$) again, and remembering that the block started from rest ($u=0$), we find:

We have our force, and we have our instantaneous velocity. In physics, mechanical power is defined as the dot product of the force vector and the velocity vector ($P=\vec{F}\cdot \vec{v}$). Because you are pushing the block in the exact same direction it's moving, the angle between the force and velocity is zero, making the dot product a simple multiplication of their magnitudes.

Let's bring it all together:

Substituting the expressions we derived:

Look closely at our final equation: $P=\frac{m{v}^2}{T^2}t$.

The terms $m$, $v$, and $T$ are all constants for this specific scenario. The only variable is time $t$. This reveals a beautiful physical truth: the instantaneous power delivered by a constant force to an object starting from rest increases linearly with time.

Even though you are pushing with the exact same force throughout the entire journey, you are doing work at a faster and faster rate as the block speeds up. If you were to plot a graph of Power versus Time, you would see a perfect straight line starting from the origin!