When the momentum of a body increases by 10%, its K.E. increases by

Options

A. 0.2
B. 0.4
C. 0.44
D. none of these

Answer

To solve the problem of how much the kinetic energy (K.E.) of a body increases when its momentum increases by 10%, we can follow these steps:

### Step 1: Understand the relationship between momentum and kinetic energy
The momentum \( P \) of a body is given by the formula:
\[ P = mv \]
where \( m \) is the mass and \( v \) is the velocity of the body.

The kinetic energy \( K.E. \) is given by:
\[ K.E. = \frac{1}{2} mv^2 \]

### Step 2: Express velocity in terms of momentum
From the momentum equation, we can express velocity \( v \) as:
\[ v = \frac{P}{m} \]

### Step 3: Substitute velocity into the kinetic energy formula
Substituting \( v \) into the kinetic energy formula gives:
\[ K.E. = \frac{1}{2} m \left(\frac{P}{m}\right)^2 = \frac{P^2}{2m} \]

### Step 4: Determine the new momentum after a 10% increase
If the momentum increases by 10%, the new momentum \( P' \) can be expressed as:
\[ P' = P + 0.1P = 1.1P \]

### Step 5: Calculate the new kinetic energy with the new momentum
Now, we can find the new kinetic energy \( K.E.' \) using the new momentum:
\[ K.E.' = \frac{(P')^2}{2m} = \frac{(1.1P)^2}{2m} = \frac{1.21P^2}{2m} \]

### Step 6: Find the increase in kinetic energy
The increase in kinetic energy \( \Delta K.E. \) is given by:
\[ \Delta K.E. = K.E.' - K.E. \]
Substituting the values we found:
\[ \Delta K.E. = \frac{1.21P^2}{2m} - \frac{P^2}{2m} \]
Factoring out \( \frac{P^2}{2m} \):
\[ \Delta K.E. = \frac{P^2}{2m} (1.21 - 1) = \frac{P^2}{2m} \cdot 0.21 \]

### Step 7: Express the increase as a percentage of the initial kinetic energy
The initial kinetic energy was:
\[ K.E. = \frac{P^2}{2m} \]
Thus, the percentage increase in kinetic energy is:
\[ \text{Percentage Increase} = \frac{\Delta K.E.}{K.E.} \times 100 = \frac{0.21 \cdot \frac{P^2}{2m}}{\frac{P^2}{2m}} \times 100 = 0.21 \times 100 = 21\% \]

### Conclusion
The kinetic energy increases by 21% when the momentum of the body increases by 10%.