Question

If momentum of an object is increased by 10% then its kinetic energy will increase by
A. 20%
B. 21%
C. 40%
D. 19%

Answer 1

Hint: To solve this question we will use the relation between kinetic energy and the momentum of a body. The relation between K.E and P is: $K.E = \dfrac{{{P^2}}}{{2m}}$, where m is the mass of the body.
Formula used: $K.E = \dfrac{{{P^2}}}{{2m}}$

Complete step-by-step solution -
We know that the relationship between the kinetic energy and momentum of a body is $K.E = \dfrac{{{P^2}}}{{2m}}$.
Given that, if momentum is increased by 10%, then by what % the K.E will increase.
So, the initial momentum be P and initial kinetic energy be $K.E = \dfrac{{{P^2}}}{{2m}}$.
Increase in momentum = 10% of P, i.e.
$ \Rightarrow \dfrac{{10}}{{100}} \times P = \dfrac{P}{{10}}$
So, final momentum = initial momentum + increased momentum.
Final momentum, ${P^1}$ = $P + \dfrac{P}{{10}} = \dfrac{{11P}}{{10}}$.
Then, final kinetic energy,
$ \Rightarrow K.{E^1} = \dfrac{{{{\left( {{P^1}} \right)}^2}}}{{2m}}$
We have final momentum = $\dfrac{{11P}}{{10}}$, putting this value, we get
$
   \Rightarrow K.{E^1} = \dfrac{{{{\left( {\dfrac{{11P}}{{10}}} \right)}^2}}}{{2m}} = \dfrac{{121{P^2}}}{{2m \times 100}} \\
   \Rightarrow K.{E^1} = \dfrac{{121}}{{100}} \times \dfrac{{{P^2}}}{{2m}} = \dfrac{{121}}{{100}}K.E \\
$.
Now, increase in kinetic energy = final kinetic energy – initial kinetic energy.
increase in kinetic energy $ = \dfrac{{121}}{{100}}K.E - K.E$
 increase in kinetic energy $ = \dfrac{{121K.E - 100K.E}}{{100}}$
increase in kinetic energy $ = \dfrac{{21K.E}}{{100}}$
% increase in kinetic energy.
\[
   \Rightarrow \dfrac{{\dfrac{{21K.E}}{{100}}}}{{K.E}} \times 100\% \\
   \Rightarrow \dfrac{{21K.E}}{{100K.E}} \times 100\% \\
   \Rightarrow 21\% \\
\]
Hence, we can see that if momentum of an object increases by 10%, then the kinetic energy will increase by 21%.
Therefore, the correct answer is option (B).

Note: Whenever we ask such questions, we first have to remember the relation between the given terms. Then we will find out the value of final momentum and then we will use that value to find the value of final kinetic energy. After that we will use the final value and initial value to find out the increase in kinetic energy. Then we can easily find out the increase %.