KE of a body is increased by `44%.` What is the percent increse in the momentum ?

Options

A. `10%`
B. `20%`
C. `30%`
D. `44%`

Answer

To solve the problem of finding the percentage increase in momentum when the kinetic energy of a body is increased by 44%, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and momentum
The kinetic energy (KE) of a body is given by the formula:
\[ KE = \frac{P^2}{2m} \]
where \( P \) is the momentum and \( m \) is the mass of the body. From this relationship, we can deduce that:
\[ KE \propto P^2 \]
This means that kinetic energy is directly proportional to the square of the momentum.

### Step 2: Express the increase in kinetic energy
Given that the kinetic energy increases by 44%, we can express the final kinetic energy \( KE_f \) as:
\[ KE_f = KE_i + 0.44 \times KE_i = 1.44 \times KE_i \]
where \( KE_i \) is the initial kinetic energy.

### Step 3: Relate the increase in kinetic energy to momentum
Since \( KE \propto P^2 \), we can write:
\[ KE_i \propto P_i^2 \]
\[ KE_f \propto P_f^2 \]
Thus, we can say:
\[ \frac{KE_f}{KE_i} = \frac{P_f^2}{P_i^2} \]

### Step 4: Substitute the values
Substituting the values we have:
\[ \frac{1.44 \times KE_i}{KE_i} = \frac{P_f^2}{P_i^2} \]
This simplifies to:
\[ 1.44 = \frac{P_f^2}{P_i^2} \]

### Step 5: Take the square root
Taking the square root of both sides gives us:
\[ \frac{P_f}{P_i} = \sqrt{1.44} \]
Calculating the square root:
\[ \sqrt{1.44} = 1.2 \]
Thus, we have:
\[ P_f = 1.2 \times P_i \]

### Step 6: Calculate the percentage increase in momentum
Now, we can find the percentage increase in momentum:
\[ \text{Percentage Increase} = \frac{P_f - P_i}{P_i} \times 100 \]
Substituting \( P_f \):
\[ \text{Percentage Increase} = \frac{(1.2 P_i - P_i)}{P_i} \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = \frac{0.2 P_i}{P_i} \times 100 = 20\% \]

### Final Answer
The percentage increase in momentum is **20%**.

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