Question

When the momentum of a body increases by 100%, its K.E. increases by:
A. 20%
B. 40%
C. 100%
D. 300%

Answer 1

Hint:To answer the above question, we need to understand the formulas of kinetic energy and momentum and find a relation between both these quantities. Kinetic energy in the energy acquired by a body due to its motion, while momentum is the product of mass and velocity of a body.

Complete step by step answer:
Kinetic energy is the type of energy that an item or particle has as a result of its movement. When work is done on an object by exerting a net force, the object accelerates and gains kinetic energy as a result. Kinetic energy is a property of a moving item or particle that is determined by its mass as well as its motion. The formula for K.E. is as follows:
$KE={\displaystyle \frac{1}{2}}m{v}^2$
Here $m$= mass of the body and $v$= velocity of the body.

The product of a particle's mass and velocity is called momentum. Momentum is a vector quantity in the sense that it has both a magnitude and a direction. The time rate of change in momentum is equal to the force applied on the particle, according to Isaac Newton's second law of motion. The formula for momentum ($p$) is:
$$p=mv$$

Comparing both equations, we get
$KE={\displaystyle \frac{p^2}{2m}}$
When momentum is increased by 100%, new kinetic energy $(K{E}^{\prime })$ becomes:
$$\begin{gathered} K{E}^{\prime }={\displaystyle \frac{{(2p)}^2}{2m}} \\ \Rightarrow K{E}^{\prime }={\displaystyle \frac{4p^2}{2m}} \end{gathered}$$
So, percentage increase in KE is,
$$\begin{gathered} {\displaystyle \frac{K{E}^{\prime }-KE}{KE}}\times 100={\displaystyle \frac{{\displaystyle \frac{4p^2-p^2}{2m}}}{{\displaystyle \frac{p^2}{2m}}}}\times 100 \\ \therefore {\displaystyle \frac{K{E}^{\prime }-KE}{KE}}\times 100=300\mathrm{\%} \end{gathered}$$

Hence,the correct answer is option D.

Note: If a constant force applies on a particle for a particular time, the product of force and time interval (the impulse) equals the change in momentum, according to Newton's second law. The momentum of a particle, on the other hand, is a measure of the time it takes for a constant force to bring it to a stop.