The kinetic energy of a body is K. If one fourth of its mass is removed and velocity is doubled, its new kinetic energy is

Options

A. K
B. 3K
C. 4K
D. `(9K)/4`

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the initial kinetic energy
The initial kinetic energy (KE) of the body is given as \( K \). The formula for kinetic energy is:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass and \( v \) is the velocity of the body.

### Step 2: Set up the initial conditions
From the problem, we know:
\[ \frac{1}{2} m v^2 = K \]
This means that the initial mass is \( m \) and the initial velocity is \( v \).

### Step 3: Determine the new mass after removing one fourth
If one fourth of the mass is removed, the new mass \( m' \) can be calculated as follows:
\[ m' = m - \frac{1}{4}m = \frac{3}{4}m \]

### Step 4: Determine the new velocity
The problem states that the velocity is doubled. Therefore, the new velocity \( v' \) is:
\[ v' = 2v \]

### Step 5: Calculate the new kinetic energy
Now, we can calculate the new kinetic energy \( KE' \) using the new mass and new velocity:
\[ KE' = \frac{1}{2} m' (v')^2 \]
Substituting the values of \( m' \) and \( v' \):
\[ KE' = \frac{1}{2} \left(\frac{3}{4}m\right) (2v)^2 \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ KE' = \frac{1}{2} \left(\frac{3}{4}m\right) (4v^2) \]
\[ KE' = \frac{3}{4}m \cdot 2v^2 \]
\[ KE' = \frac{3}{2} mv^2 \]

### Step 7: Relate the new kinetic energy to the initial kinetic energy
Recall that the initial kinetic energy \( K \) is:
\[ K = \frac{1}{2} mv^2 \]
Thus, we can express \( KE' \) in terms of \( K \):
\[ KE' = 3K \]

### Final Answer
The new kinetic energy after removing one fourth of the mass and doubling the velocity is:
\[ \text{New Kinetic Energy} = 3K \]

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