If the momentum of a body decreases by `30%`, then kinetic energy decreases by

Options

A. `60%`
B. `51%`
C. `69%`
D. `90%`

Explanation

To solve the problem of how much the kinetic energy decreases when the momentum of a body decreases by 30%, 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. The kinetic energy \( K \) is given by:
\[ K = \frac{1}{2} mv^2 \]

### Step 2: Express kinetic energy in terms of momentum
We can rewrite the kinetic energy in terms of momentum:
\[ K = \frac{1}{2} mv^2 = \frac{1}{2} m \left(\frac{P}{m}\right)^2 = \frac{P^2}{2m} \]
This shows that kinetic energy is proportional to the square of the momentum.

### Step 3: Determine the new momentum after a 30% decrease
If the momentum decreases by 30%, the new momentum \( P_2 \) can be expressed as:
\[ P_2 = P_1 - 0.3P_1 = 0.7P_1 \]

### Step 4: Calculate the new kinetic energy
Using the relationship derived earlier, we can express the new kinetic energy \( K_2 \):
\[ K_2 = \frac{P_2^2}{2m} = \frac{(0.7P_1)^2}{2m} = \frac{0.49P_1^2}{2m} \]

### Step 5: Relate the new kinetic energy to the original kinetic energy
The original kinetic energy \( K_1 \) is:
\[ K_1 = \frac{P_1^2}{2m} \]
Now, we can find the ratio of the new kinetic energy to the original kinetic energy:
\[ \frac{K_2}{K_1} = \frac{0.49P_1^2}{2m} \div \frac{P_1^2}{2m} = 0.49 \]
Thus, we find:
\[ K_2 = 0.49 K_1 \]

### Step 6: Calculate the percentage decrease in kinetic energy
The percentage decrease in kinetic energy can be calculated as:
\[ \text{Percentage Decrease} = \frac{K_1 - K_2}{K_1} \times 100 \]
Substituting \( K_2 \):
\[ \text{Percentage Decrease} = \frac{K_1 - 0.49K_1}{K_1} \times 100 = \frac{0.51K_1}{K_1} \times 100 = 51\% \]

### Final Answer
The kinetic energy decreases by **51%**.

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