Question

The expression for kinetic energy is:
$
  A.{\text{ }}\dfrac{1}{2}mv \\
  B.{\text{ }}\dfrac{1}{3}m{v^2} \\
  C.{\text{ }}\dfrac{1}{{2m}}{v^2} \\
  D.{\text{ }}\dfrac{1}{2}m{v^2} \\
 $

Answer 1

Hint: Kinetic energy is directly proportional to the mass of the object and half of the square of velocity. Greater the motion i.e. greater the velocity and greater the mass, larger is the kinetic energy of the body.

Formulas used:
$
  F = ma{\text{ (Newton's second law of motion)}} \\
  {{\text{v}}^2} - {u^2} = 2ad{\text{ }} \\
  W = F \times d \\
$

Complete step by step answer:

A body can possess energy by the virtue of its position as well as by the virtue of its motion. The former constitutes what is called the potential energy of the body whereas the latter constitutes the kinetic energy. Therefore, we can define kinetic energy as the energy possessed by a moving body because of its motion. Every moving body has an amount of kinetic energy associated with it.

To make a body move, we need to apply some force on it. The body gets accelerated by the force applied and hence some work is being done on it. According to the law of conservation of energy, energy can neither be created nor destroyed. Therefore, the energy used in doing work on the body is converted into kinetic energy and the body starts moving. This energy which makes the body move is called the kinetic energy.

Consider a body of mass m initially at rest, is being pushed with force F and it travels a distance d.
$
  \therefore u = 0 \\
   \Rightarrow {v^2} = 2ad \\
 $

The work done on the body is given as

$ W = F \times d \\
   \Rightarrow W = m \times d \times \dfrac{{{v^2}}}{{2d}} \\
   \Rightarrow W = \dfrac{{m{v^2}}}{2} \\
  {\text{Since, K}}{\text{.E}}{\text{. = Work done}} \\
  \therefore K.E. = \dfrac{1}{2}m{v^2} \\ $

Hence, the correct answer is option D. Therefore, we can say that the kinetic energy of a body is directly proportional to its mass and half the square of velocity.

Additional Information:
The total energy of a body is the sum of its kinetic energy and potential energy. A body at rest has zero kinetic energy and non-zero potential energy.

Note: The student might make a mistake of choosing option A because here also there is direct dependence of kinetic energy with mass and velocity. But kinetic energy depends on the square of velocity. This fact should be remembered. Also, there is always a factor of ½ associated with kinetic energy so students should not make the mistake of choosing option B. Similarly, option C is also wrong, as K.E. is directly proportional to mass.