Momentum of a body increae by `0.02%` percent increase in KE is

Options

A. `0.02%`
B. `0.04%`
C. `0.01%`
D. `0.08%`

Answer

To solve the problem of finding the percentage increase in kinetic energy when the momentum of a body increases by 0.02%, we can follow these steps:

### Step 1: Understand the relationship between momentum and kinetic energy
The kinetic energy (KE) of a body is related to its momentum (p) by the formula:
\[ KE = \frac{p^2}{2m} \]
where \(m\) is the mass of the body.

### Step 2: Define the percentage change in momentum
Given that the momentum increases by 0.02%, we can express this mathematically as:
\[ \frac{\Delta p}{p} \times 100 = 0.02\% \]
where \(\Delta p\) is the change in momentum and \(p\) is the initial momentum.

### Step 3: Relate the change in kinetic energy to the change in momentum
To find the percentage change in kinetic energy, we can differentiate the kinetic energy formula with respect to momentum. The percentage change in kinetic energy can be expressed as:
\[ \frac{\Delta KE}{KE} \times 100 = 2 \times \frac{\Delta p}{p} \times 100 \]
This is derived from the fact that kinetic energy is proportional to the square of momentum.

### Step 4: Substitute the percentage change in momentum
Now, substituting the percentage change in momentum (0.02%) into the equation:
\[ \frac{\Delta KE}{KE} \times 100 = 2 \times 0.02\% \]

### Step 5: Calculate the percentage change in kinetic energy
Calculating the right-hand side:
\[ \frac{\Delta KE}{KE} \times 100 = 0.04\% \]

### Conclusion
Thus, the percentage increase in kinetic energy when the momentum increases by 0.02% is:
\[ \text{Percentage increase in KE} = 0.04\% \]

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