Question

Johnny and his sister Jane race up a hill. Johnny weighs twice as much as Jane and takes twice as long as Jane to reach the top. Compared to Jane:
A. Johnny did more work and delivered more power.
B. Johnny did more work and delivered the same power.
C. Johnny did more work and delivered less power.
D. Johnny did less work and delivered less power.

Answer 1

Recall that the work done is the displacement of any object under the influence of a force. Here, the force under consideration is the gravitational force, and the work done is against the acceleration due to gravity. Using this, determine the work done by both Johnny and Jane. Following this, we know that the power is the rate at which this work is done and is dependent on the time taken to do this work. Using this determines the power of both Johnny and Jane with which they race up the hill and obtain the appropriate solution from their comparison.

Formula Used:
Work done $W = mgh$
Power $P = \dfrac{W}{t}$

Complete Solution:
We have two persons Johnny and Jane
with weights $w_{johnny} = 2w_{jane}$, and
the time they take to climb up the stairs is $t_{johnny} = 2t_{jane}$

(i)The work done by both the persons is against the gravitational force arising from their weight to race up the hill of height say h.
We know that $F_{gravitational} = mg = w$, where m is the mass, g is the acceleration due to gravity and w is the weight.
$W_{johnny} = F_{gravitational}h = w_{johnny}h = 2w_{jane}h$
$W_{jane} = F_{gravitational}h = w_{jane}h$
$\Rightarrow W_{johnny} = 2W_{jane}$
Therefore, compared to Jane, Johnny does more work.

(ii)The power delivered by Johnny and Jane in racing up the hill is defined as the rate at which the work is done.
$P_{johnny} = \dfrac{W_{johnny}}{t_{johnny}} = \dfrac{2W_{jane}}{2t_{jane}} = \dfrac{W_{jane}}{t_{jane}}$
$P_{jane} = \dfrac{W_{jane}}{t_{jane}}$
$\Rightarrow P_{johnny} = P_{jane}$
Therefore, compared to Jane, Johnny delivers the same amount of power.

Thus, the correct choice would be B. Johnny did more work and delivered the same power.

Note:
Do not get confused between work, energy and power.
Work is said to be done when a force applied changes the state of rest or motion of an object. The SI unit of work is joules (J). It is usually given as:
$W = F.d$, where F is the force applied and d is the resulting displacement produced.
Energy of a body is defined as its capacity to do work or bring about a change in the state of the body. Its SI unit is joules (J).
Finally, power is defined as the rate at which the work is done or the rate at which the energy conversion from one form to the other takes place. Its SI unit is watts (W), and is usually defined as:
$P = \dfrac{W}{t}$, where t is the time taken to do the work.