Question

Kinetic energy of a body is directly proportional to it is:
A. weight
B. Speed
C. Mass
D. none

Answer 1

As a first step, you could recall the expression for kinetic energy. In case if you don’t remember you could easily derive it by equating it to the work done in bringing the body to rest by applying a retarding force. Thus, from the expression, you could easily find to which among the given physical quantities is the kinetic energy of the body directly proportional.

Formula used:
Work done,
$W=F.S$
Newton’s second law,
$F=ma$
Newton’s equation of motion,
${{v}^{2}}-{{u}^{2}}=2as$

Complete answer:
In the question, we are asked to find which among the given physical quantities in the given options is the kinetic energy directly proportional.
In order to answer this, we have to know the expression for kinetic energy. We could derive this relation very easily. We know that kinetic energy is equal to the work done by the retarding force to stop it and work done is the product of force and displacement. So,
$K.E=W=F.S$ ………………………………………….. (1)
But, by Newton’s second law,
$F=ma$ …………………………………. (2)
Let the body’s initial velocity be v and its final velocity will be 0 as it is going to rest. We have the equation of motion given by,
${{v}^{2}}-{{u}^{2}}=2as$
$\Rightarrow {{0}^{2}}-{{v}^{2}}=2\left( -a \right)S$
$\therefore S=\dfrac{{{v}^{2}}}{2a}$ …………………………………… (3)
Substituting (2) and (3) in (1), we get,
$K.E=ma\times \dfrac{{{v}^{2}}}{2a}$
Therefore, we get the expression for Kinetic energy as,
$K.E=\dfrac{1}{2}m{{v}^{2}}$
Therefore, we find that kinetic energy of the body is directly proportional to the mass and the square of its velocity.

Hence, option C is found to be the right answer.

Note:
While substituting the quantities in the equation of motion, we have substituted negative acceleration because the body is undergoing deceleration here. Also, from the expression for kinetic energy we see that with the increase in velocity (speed) the kinetic energy of the body increases. However, we cannot say that the body is directly proportional to the speed as it is directly proportional to the square of the speed of the body.