If momentum is increased by 20% , then K.E. increase by

Options

A. 0.55
B. 0.77
C. 0.66
D. 0.44

Explanation

To solve the problem of how much the kinetic energy (K.E.) increases when momentum is increased by 20%, we can follow these steps:

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

### Step 2: Define initial momentum and kinetic energy
Let the initial momentum be \( p \) and the initial kinetic energy be \( K.E. = \frac{p^2}{2m} \).

### Step 3: Calculate the new momentum after a 20% increase
If the momentum is increased by 20%, the new momentum \( p' \) can be expressed as:
\[ p' = p + 0.2p = 1.2p \]

### Step 4: Calculate the new kinetic energy with the new momentum
Now, we can find the new kinetic energy \( K.E.' \) using the new momentum:
\[ K.E.' = \frac{(p')^2}{2m} = \frac{(1.2p)^2}{2m} = \frac{1.44p^2}{2m} \]

### Step 5: Determine the change in kinetic energy
The change in kinetic energy \( \Delta K.E. \) is given by:
\[ \Delta K.E. = K.E.' - K.E. = \frac{1.44p^2}{2m} - \frac{p^2}{2m} \]
\[ \Delta K.E. = \frac{1.44p^2 - p^2}{2m} = \frac{0.44p^2}{2m} \]

### Step 6: Express the change in kinetic energy in terms of the initial kinetic energy
Since the initial kinetic energy \( K.E. = \frac{p^2}{2m} \), we can substitute this into our equation:
\[ \Delta K.E. = 0.44 \cdot K.E. \]

### Step 7: Calculate the percentage increase in kinetic energy
To find the percentage increase in kinetic energy:
\[ \text{Percentage Increase} = \left( \frac{\Delta K.E.}{K.E.} \right) \times 100 = \left( \frac{0.44K.E.}{K.E.} \right) \times 100 = 44\% \]

### Conclusion
Thus, if momentum is increased by 20%, the kinetic energy increases by **44%**.

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