If the momentum of a body is increased by 100%, then the percentage increase in the kinetic energy is

Options

A. 1.5
B. 2
C. 2.25
D. 3

Answer

To solve the problem of determining the percentage increase in kinetic energy when the momentum of a body is increased by 100%, we can follow these steps:

### Step 1: Understand the relationship between momentum and kinetic energy
Momentum (P) is defined as:
\[ P = mv \]
where \( m \) is the mass of the body and \( v \) is its velocity.

Kinetic energy (KE) is defined as:
\[ KE = \frac{1}{2} mv^2 \]

### Step 2: Define initial momentum and kinetic energy
Let the initial momentum be \( P_i = mv \) and the initial kinetic energy be:
\[ KE_i = \frac{1}{2} mv^2 \]

### Step 3: Calculate the final momentum after a 100% increase
If the momentum is increased by 100%, the final momentum \( P_f \) becomes:
\[ P_f = P_i + 100\% \text{ of } P_i = P_i + P_i = 2P_i \]

### Step 4: Relate final momentum to final velocity
Since \( P_f = mv_f \), we have:
\[ 2P_i = mv_f \]
Substituting \( P_i = mv \) gives:
\[ 2(mv) = mv_f \]
Thus, we can cancel \( m \) (assuming \( m \neq 0 \)):
\[ v_f = 2v \]

### Step 5: Calculate the final kinetic energy
Now, we can find the final kinetic energy \( KE_f \):
\[ KE_f = \frac{1}{2} mv_f^2 = \frac{1}{2} m(2v)^2 = \frac{1}{2} m(4v^2) = 2mv^2 \]

### Step 6: Calculate the percentage increase in kinetic energy
Now, we find the change in kinetic energy:
\[ \Delta KE = KE_f - KE_i = 2mv^2 - \frac{1}{2} mv^2 = 2mv^2 - 0.5mv^2 = 1.5mv^2 \]

The percentage increase in kinetic energy is given by:
\[ \text{Percentage Increase} = \left( \frac{\Delta KE}{KE_i} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{1.5mv^2}{\frac{1}{2} mv^2} \right) \times 100 = \left( \frac{1.5}{0.5} \right) \times 100 = 3 \times 100 = 300\% \]

### Final Answer
The percentage increase in kinetic energy when the momentum of a body is increased by 100% is **300%**.

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