Question

A man of mass $50kg$ is standing on a $100kg$ plank kept on a frictionless horizontal floor, initially both are at rest. If the man starts walking on the plank with speed $6m/s$ towards right relative to the plank, then amount of muscle energy spent by the man is?​

Answer 1

Hint: To discuss relative motion in one or more dimensions, we need to introduce the concept of reference frame. When we say that an object has a certain velocity, we must state that its velocity with respect to a given reference frame.

Complete step by step solution:
Let Planck's velocity v be on the left side, so man's velocity is right (6-v) on the right. according to the conservation of linear motion
Initial momentum = final momentum
100v = 50 (6-v)
Therefore, v = 2m / sec
And velocity of man = 6-2 = 4 m/sec
Also work gain in kinetic energy is equal to the Muscle energy spent
$\Rightarrow$ Muscle energy spent is equal to:
$\Rightarrow$ $\dfrac{1}{2} \times 100 \times {2^2}{\text{ + }}\dfrac{1}{2} \times 50 \times {4^2}{\text{ }}$
$\therefore$ Muscle energy spent = $600J$.

Additional Information:
We encounter occasions where one or more objects move in a frame that is non-stationary with respect to another observer.
For example, a boat crosses a river that is flowing at some rate or an airplane is facing the wind during its speed. In all such examples, to describe the absolute motion of the object, we have to consider the effect that the medium is causing on the object. While doing so, we calculate the relative velocity of the object, considering the velocity of the particle as well as the velocity of the medium. Here, we will learn how to calculate relative velocity.

Note: Relative velocity can be negative, as the relative velocity is the difference between the two velocities, regardless of the magnitude of the velocities, it can be negative. The need to use relative velocity is that it is used to differentiate when the object is at rest or moving.