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

Solution & Explanation

1. Calculate the Constant Acceleration

The body starts from rest, which means its initial velocity ($$u$$) is zero:

$$u = 0 \,\, \text{m/s}$$

According to the first equation of motion ($$v = u + a \cdot T$$), the body reaches a speed of $$v$$ in a total time interval $$T$$. Substituting these values allows us to find the uniform acceleration ($$a$$):

$$v = 0 + a \cdot T$$

$$a = \frac{v}{T}$$

2. Express Instantaneous Velocity as a Function of Time

Now, let us determine the velocity ($$v_t$$) of the body at any arbitrary, instantaneous moment of time $$t$$ during its acceleration phase:

$$v_t = u + a \cdot t$$

$$v_t = 0 + \left(\frac{v}{T}\right) \cdot t$$

$$v_t = \frac{v \cdot t}{T}$$

3. Compute Instantaneous Power ($$P$$)

Instantaneous power delivered to a moving body is defined as the product of the net force ($$F$$) acting on it and its instantaneous velocity ($$v_t$$):

$$P = F \cdot v_t$$

According to Newton's Second Law of Motion, the constant horizontal force is the product of mass ($$m$$) and its uniform acceleration ($$a$$):

$$F = m \cdot a = m \cdot \left(\frac{v}{T}\right)$$

Substituting both the force ($$F$$) and instantaneous velocity ($$v_t$$) expressions into the power formula yields:

$$P = \left( m \cdot \frac{v}{T} \right) \cdot \left( \frac{v \cdot t}{T} \right)$$

$$P = \frac{m \cdot v^2}{T^2} \cdot t$$

Concept Check: Because the acceleration is uniform, the force driving the object remains completely constant. However, as the body gains speed over time, more power must be pumped into the system to maintain that constant acceleration, causing the instantaneous power to scale linearly with time ($$P \propto t$$).

Correct Option Key: Option A ($$\frac{m \cdot v^2}{T^2} \cdot t$$)