Momentum of a particle is increased by `50%`. By how much percentage kinetic energy of particle will increase?

Explanation

To solve the problem of how much the kinetic energy of a particle increases when its momentum is increased by 50%, we can follow these steps:

### Step 1: Understand the relationship between momentum and kinetic energy
The kinetic energy (KE) of a particle is given by the formula:
\[ KE = \frac{p^2}{2m} \]
where \( p \) is the momentum and \( m \) is the mass of the particle.

### Step 2: Define the initial momentum
Let the initial momentum of the particle be \( p \). Therefore, the initial kinetic energy \( KE_i \) can be expressed as:
\[ KE_i = \frac{p^2}{2m} \]

### Step 3: Calculate the new momentum after a 50% increase
If the momentum is increased by 50%, the new momentum \( p' \) becomes:
\[ p' = p + 0.5p = 1.5p \]

### Step 4: Calculate the final kinetic energy
Using the new momentum, we can calculate the final kinetic energy \( KE_f \):
\[ KE_f = \frac{(p')^2}{2m} = \frac{(1.5p)^2}{2m} = \frac{2.25p^2}{2m} \]

### Step 5: Simplify the final kinetic energy
This simplifies to:
\[ KE_f = \frac{2.25p^2}{2m} = 1.125 \cdot \frac{p^2}{2m} = 1.125 \cdot KE_i \]

### Step 6: Calculate the percentage increase in kinetic energy
The percentage increase in kinetic energy can be calculated using the formula:
\[ \text{Percentage Increase} = \frac{KE_f - KE_i}{KE_i} \times 100 \]
Substituting the values we found:
\[ \text{Percentage Increase} = \frac{1.125 \cdot KE_i - KE_i}{KE_i} \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = \frac{0.125 \cdot KE_i}{KE_i} \times 100 = 12.5\% \]

### Final Answer
Thus, the percentage increase in the kinetic energy of the particle is **125%**.

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