Question

The kinetic energy of a body increases by 125%. The percentage increase in its momentum is:
1. 50%
2. 62.5%
3. 250%
4. 200%

Answer 1

Hint: First, we calculate the final kinetic energy by adding 125% to initial kinetic energy. Then, we express the initial and final kinetic energy in terms of momentum. The final momentum is expressed in terms of initial momentum and then the percentage increase is calculated.

Formula used: Kinetic energy is given by,
\[KE=\dfrac{{{p}^{2}}}{2m}\]
where, \[p\] is momentum and m is mass of the body.
Percentage increase is calculated as:
\[\dfrac{\text{Final value }-\text{ initial value}}{\text{initial value}}\times 100\]

Complete step by step answer:
Given,
The kinetic energy of a body increases by 125%, therefore,
\[K{{E}_{f}}=125%\text{ of }K{{E}_{i}}+K{{E}_{i}}\]
where, \[K{{E}_{f}}\] is final kinetic energy and \[K{{E}_{i}}\] is initial kinetic energy.
Simplifying, we get,
\[\begin{aligned}
  & K{{E}_{f}}=\dfrac{125}{100}\text{ of }K{{E}_{i}}+K{{E}_{i}} \\
 &\Rightarrow K{{E}_{f}}=\dfrac{125}{100}\text{ of }K{{E}_{i}}+K{{E}_{i}} \\
 &\Rightarrow K{{E}_{f}}=\dfrac{5}{4}K{{E}_{i}}+K{{E}_{i}} \\
 &\Rightarrow K{{E}_{f}}=\dfrac{9}{4}K{{E}_{i}} \\
\end{aligned}\]
Substituting \[KE=\dfrac{{{p}^{2}}}{2m}\] in the above results, we get,
\[\dfrac{{{p}_{f}}^{2}}{2m}=\dfrac{9}{4}\dfrac{{{p}_{i}}^{2}}{2m}\]
where, \[{{p}_{f}}\] is final momentum and \[{{p}_{i}}\] is initial momentum.
Cancelling out 2m on both sides, we get,
\[{{p}_{f}}^{2}=\dfrac{9}{4}{{p}_{i}}^{2}\]
We find square root of both sides, therefore,
\[{{p}_{f}}=\dfrac{3}{2}{{p}_{i}}\]
Percentage increase in momentum is calculated as:
\[\dfrac{{{p}_{f}}-{{p}_{i}}}{{{p}_{i}}}\times 100\]
On substituting the values and simplifying, we get,
\[\begin{aligned}
  & \dfrac{\dfrac{3}{2}{{p}_{i}}-{{p}_{i}}}{{{p}_{i}}}\times 100 \\
 & =\dfrac{\dfrac{1}{2}{{p}_{i}}}{{{p}_{i}}}\times 100 \\
 & =\dfrac{1}{2}\times 100 \\
 &\ =50\% \\
\end{aligned}\]

So, the correct answer is “Option 1”.

Note: It should be kept in mind that the initial and the final masses will be the same. Therefore, when expressing the initial and the final kinetic energy in terms of momentum, only momentum will be taken as initial and final and not the mass. Mass remains constant. It can be noted that the increase in momentum is not as great as the increase in kinetic energy.