If the kinetic energy of a body increases by `0.1 %` the percent increase of its momentum will be

Options

A. (a) `0.05%`
B. (b) `0.1%`
C. (c) `1.0%`
D. (d) `10%`

Explanation

To solve the problem, we need to find the percent increase in momentum when the kinetic energy of a body increases by 0.1%. Let's go through the steps systematically.

### Step 1: Define the initial kinetic energy
Let the initial kinetic energy of the body be represented as \( K \).

### Step 2: Calculate the final kinetic energy
The kinetic energy increases by 0.1%. Therefore, the final kinetic energy \( K' \) can be expressed as:
\[ K' = K + 0.1\% \text{ of } K = K + \frac{0.1}{100} K = K \left(1 + \frac{0.1}{100}\right) = K \left(1 + 0.001\right) = K \times 1.001 \]

### Step 3: Express the initial momentum
The momentum \( P_i \) of the body can be expressed in terms of kinetic energy:
\[ P_i = \sqrt{2mK} \]
where \( m \) is the mass of the body.

### Step 4: Express the final momentum
Similarly, the final momentum \( P_f \) can be expressed as:
\[ P_f = \sqrt{2mK'} = \sqrt{2m \left(K \times 1.001\right)} = \sqrt{2mK \times 1.001} = \sqrt{2mK} \sqrt{1.001} = P_i \sqrt{1.001} \]

### Step 5: Calculate the percent increase in momentum
The percent increase in momentum can be calculated using the formula:
\[ \text{Percent Increase} = \left(\frac{P_f - P_i}{P_i}\right) \times 100 \]
Substituting \( P_f \):
\[ \text{Percent Increase} = \left(\frac{P_i \sqrt{1.001} - P_i}{P_i}\right) \times 100 = \left(\sqrt{1.001} - 1\right) \times 100 \]

### Step 6: Approximate \( \sqrt{1.001} \)
Using the binomial approximation for small values (since \( 0.001 \) is small):
\[ \sqrt{1+x} \approx 1 + \frac{x}{2} \quad \text{for small } x \]
Thus, for \( x = 0.001 \):
\[ \sqrt{1.001} \approx 1 + \frac{0.001}{2} = 1 + 0.0005 = 1.0005 \]

### Step 7: Calculate the percent increase
Now substituting back:
\[ \text{Percent Increase} = \left(1.0005 - 1\right) \times 100 = 0.0005 \times 100 = 0.05\% \]

### Conclusion
The percent increase in momentum when the kinetic energy increases by 0.1% is \( 0.05\% \).

### Final Answer
Thus, the correct option is **A: 0.05%**.

---