Question

A body of mass $m$ accelerates uniformly from rest to $v_1$ in time $t_1$ . As a function of time $t$, find the instantaneous power delivered to the body.

Answer 1

Step 1: Given data

Given that:-

A mass $m$ acquiring velocity $v_1$from rest in time $t_1$

Step 2: Using equation

$\Rightarrow v=u+at$

where $u$ is the initial velocity,

Step 3: Calculate the power

$$\begin{gathered} \Rightarrow v_1=0+a{t}_1 \\ \Rightarrow a=\frac{v_1}{t_1} \end{gathered}$$

Let at any instant $\text{'}t\text{'}$ the velocity of the body $\text{'}v\text{'}$' is

$$\begin{gathered} \Rightarrow v=0+at \\ \Rightarrow v=at \end{gathered}$$

According to Newton's second law, the force acting on the body in terms of acceleration and mass is,

$F=ma$

The instantaneous power is given by

$$\begin{gathered} \Rightarrow P=Fv \\ \Rightarrow P=mav \\ \Rightarrow P=m{a}^{2}t \\ \Rightarrow P=\frac{m{v}_1^2}{t_1^2}t \end{gathered}$$

Hence, the instantaneous power delivered to the body is $\frac{m{v}_1^2}{t_1^2}t$.