If the kinetic energy of a body increases by 125% the percentage increase in its momentum is

Options

A. 0.5
B. 0.625
C. 2.5
D. 2

Answer

To solve the problem of finding the percentage increase in momentum when the kinetic energy of a body increases by 125%, we can follow these steps:

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

### Step 2: Express the increase in kinetic energy
If the kinetic energy increases by 125%, it means the new kinetic energy \( K.E._f \) can be expressed as:
\[ K.E._f = K.E._i + 1.25 \times K.E._i = 2.25 \times K.E._i \]
where \( K.E._i \) is the initial kinetic energy.

### Step 3: Relate the change in kinetic energy to momentum
Using the relationship between kinetic energy and momentum, we can express the initial and final kinetic energies in terms of momentum:
- Initial kinetic energy:
\[ K.E._i = \frac{p_i^2}{2m} \]
- Final kinetic energy:
\[ K.E._f = \frac{p_f^2}{2m} \]

### Step 4: Set up the equation
From the above expressions, we have:
\[ 2.25 \times K.E._i = K.E._f \]
Substituting the expressions for kinetic energy, we get:
\[ 2.25 \times \frac{p_i^2}{2m} = \frac{p_f^2}{2m} \]

### Step 5: Simplify the equation
Since \( 2m \) is a common factor, we can cancel it out:
\[ 2.25 p_i^2 = p_f^2 \]

### Step 6: Solve for the final momentum
Taking the square root of both sides:
\[ p_f = \sqrt{2.25} \times p_i = 1.5 \times p_i \]

### Step 7: Calculate the change in momentum
The change in momentum \( \Delta p \) is given by:
\[ \Delta p = p_f - p_i = (1.5 p_i - p_i) = 0.5 p_i \]

### Step 8: Calculate the percentage increase in momentum
The percentage increase in momentum can be calculated as:
\[ \text{Percentage Increase} = \left( \frac{\Delta p}{p_i} \right) \times 100 = \left( \frac{0.5 p_i}{p_i} \right) \times 100 = 50\% \]

### Final Answer
The percentage increase in momentum is **50%**.

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