Question

An elevator is moving upwards with an acceleration a it has velocity v when a man inside the elevator lifts a body of mass m through a height h (in time t). The average power developed by the man is:
A) $\dfrac{{m\left( {a + g} \right)h}}{t}$.
B) $m\left( {a + g} \right)\left( {v + \dfrac{1}{2}a{t^2}} \right)$
C) $\dfrac{{mgh}}{t}$
D) $mg\left( {v + \dfrac{1}{2}a{t^2}} \right)$

Answer 1

The body travelling with an acceleration a and the motion of the direction is towards the ground then the total acceleration on the body of certain mass will be the addition of the acceleration a and the acceleration due to gravity i.e. g.

Formula used: The formula of the power is given by,
$P = F \times v$
Where power is P force is F velocity is v.
The formula of force is given by,
$F = m \times a$
Where force is F mass is m and acceleration is a.

Complete step by step answer:
It is given in the problem that the an elevator is moving upwards with an acceleration a it has velocity v at this instant the man inside the elevator lifts a body of mass m through a height h and the time taken is t. we need to find the average power developed by the man.
The mass of the body is m and the acceleration with which the body is accelerating is $a + g$. The force applied by the man on the body of mass m should be equal to,
$\Rightarrow F = m \times a$
$\Rightarrow F = m \times \left( {a + g} \right)$………eq. (1)
The power developed by the man to lift the body of mass m is given by,
$\Rightarrow P = F \times v$………eq. (2)
Replace the value of force from equation (1) to equation (2).
$\Rightarrow P = F \times v$
$\Rightarrow P = m \times \left( {a + g} \right) \times v$………eq. (3)
The velocity is equal to the ratio of distance and time taken t.
The distance travelled by the body is h and time taken is t.
$v = \dfrac{h}{t}$
Replacing the value of v in the equation (3) we get,
$\Rightarrow P = m \times \left( {a + g} \right) \times v$
$\Rightarrow P = m \times \left( {a + g} \right) \cdot \left( {\dfrac{h}{t}} \right)$
$\Rightarrow P = \dfrac{{m \cdot \left( {a + g} \right) \cdot h}}{t}$
The power developed by the man is equal to $P = \dfrac{{m \cdot \left( {a + g} \right) \cdot h}}{t}$. The correct answer for this problem is option A.

Note: The man is inside the elevator and the body is also in the same frame therefore the acceleration that the man develops will be the same as the acceleration produced by the body of mass m. The power is the rate of change of force with respect to time.