Question

A body is initially at rest. It undergoes one dimensional motion with constant acceleration. Show that the power delivered to it at time ‘t’ is proportional to ‘t’.

Answer 1

Here the motion is one-dimensional motion, so the body is moving in a straight line. The body starts from rest, so its initial velocity is zero. The acceleration is constant. It means the body changes its velocity and the change is uniform equal in equal interval of time. We know power is defined as the rate of doing work.

Complete step by step answer:
Let us assume the acceleration of the body ‘a’. Power can be written as, $P=\dfrac{W}{t}$,
where W is the work and t is time taken to do that work. Rewriting it,
P=\dfrac{W}{t} \\\ \Rightarrow P=\dfrac{Fd}{t} \\\ \Rightarrow P=F\dfrac{d}{t} \\\ \Rightarrow v=\dfrac{d}{t} \\\ \Rightarrow P=Fv \\\
Now we use first equation of motion: $v=u+at$
Since the body starts from rest, so u= 0, $\therefore v=at$
$P=Fat \\\ \therefore P\propto t$

Hence, proved.

Additional Information:
We can define power as the rate of doing work, it is the work done in unit time. The SI unit of power is Watt (W) which is joules per second (J/s). Sometimes the power of motor vehicles and other machines is given in terms of Horsepower (hp), which is approximately equal to 745.7 watts.We can define average power as the total energy consumed divided by the total time taken. In simple language, we can say that average power is the average amount of work done or energy converted per unit of time.

Note: watt (W), is the SI unit of power and it is defined as the rate of doing 1 J of work per second. P=Fv, in this equation we calculate the power is the instantaneous power. we are calculating in this problem not just power but the instantaneous power, because we had taken the velocity at a given instant and then we had used the equation to modify to write it in terms of velocity.