If momentum of a body is increased by 20%, then its kinetic energy increases by
- 36%
- 40%
- 44%
- 48%
The Correct Option is C
Approach Solution - 1
To solve this problem, we need to understand the relationship between momentum and kinetic energy. Let's break it down step-by-step:
1. **Formula for Momentum and Kinetic Energy**: The momentum (\(p\)) of a body is given by the formula:
\(p = mv\)
where \(m\) is the mass of the body and \(v\) is its velocity.
The kinetic energy (\(KE\)) of a body is given by the formula:
\(\text{KE} = \frac{1}{2}mv^2\)
2. **Relating Momentum and Kinetic Energy**: We can express kinetic energy in terms of momentum. By substituting \(v = \frac{p}{m}\) (from the momentum formula) into the kinetic energy equation, we have:
\(\text{KE} = \frac{1}{2}m\left(\frac{p}{m}\right)^2 = \frac{p^2}{2m}\)
3. **Calculate the Increase in Kinetic Energy**: Given that the momentum of the body increases by 20%, the new momentum is:
\(p_{\text{new}} = p + 0.2p = 1.2p\)
The new kinetic energy based on the increased momentum is:
\(\text{KE}_{\text{new}} = \frac{{(1.2p)}^2}{2m} = \frac{1.44p^2}{2m}\)
4. **Determine the Percentage Increase in Kinetic Energy**: The initial kinetic energy was \(\frac{p^2}{2m}\). The percentage increase in kinetic energy can be calculated as:
\(\text{Percentage Increase} = \left(\frac{\text{KE}_{\text{new}} - \text{KE}}{\text{KE}}\right) \times 100%\)
\(= \left(\frac{\frac{1.44p^2}{2m} - \frac{p^2}{2m}}{\frac{p^2}{2m}}\right) \times 100%\)
\(= \left(\frac{1.44p^2 - p^2}{p^2}\right) \times 100%\)
\(= \left(0.44\right) \times 100%\)
\(= 44%\)
Hence, when the momentum of a body is increased by 20%, its kinetic energy increases by 44%.
Approach Solution -2
\(K=\frac{p^2}{2m}\)
\(K=\frac{(1.2p)^2}{2m}\)
\(⇒\frac{K−K}{K}=(1.2)^2−1=0.44\)
⇒ Kinetic energy increases by 44%
So, the correct option is (C): 44%
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Concepts Used:
Kinetic energy
Kinetic energy of an object is the measure of the work it does as a result of its motion. Kinetic energy is the type of energy that an object or particle has as a result of its movement. When an object is subjected to a net force, it accelerates and gains kinetic energy as a result. Kinetic energy is a property of a moving object or particle defined by both its mass and its velocity. Any combination of motions is possible, including translation (moving along a route from one spot to another), rotation around an axis, vibration, and any combination of motions.
