If the units of M and L are doubled then the unit of kinetic energy will become

Options

A. 8 times
B. 16 times
C. 4 times
D. 2 times

Explanation

To solve the problem of how the unit of kinetic energy changes when the units of mass (M) and length (L) are doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for kinetic energy (KE)**:
The formula for kinetic energy is given by:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass and \( v \) is the velocity.

2. **Express velocity in terms of length and time**:
Velocity \( v \) is defined as displacement (length) divided by time:
\[ v = \frac{L}{T} \]
Therefore, the square of velocity can be expressed as:
\[ v^2 = \left(\frac{L}{T}\right)^2 = \frac{L^2}{T^2} \]

3. **Substitute the expression for velocity into the kinetic energy formula**:
Now substituting \( v^2 \) into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\frac{L^2}{T^2}\right) = \frac{1}{2} m \frac{L^2}{T^2} \]

4. **Identify the dimensions of kinetic energy**:
The dimensional formula for kinetic energy can be expressed as:
\[ [KE] = [M][L^2][T^{-2}] = M L^2 T^{-2} \]

5. **Consider the effect of doubling the units of M and L**:
If the units of mass (M) and length (L) are doubled, we can express the new units as:
- New mass \( M' = 2M \)
- New length \( L' = 2L \)

6. **Substitute the new units into the kinetic energy formula**:
The new kinetic energy will be:
\[ KE' = \frac{1}{2} (2M) \left(\frac{(2L)^2}{T^2}\right) \]
Simplifying this gives:
\[ KE' = \frac{1}{2} (2M) \left(\frac{4L^2}{T^2}\right) = \frac{1}{2} \cdot 2 \cdot 4 \cdot \frac{M L^2}{T^2} = 4 \cdot \frac{M L^2}{T^2} \]

7. **Calculate the factor by which kinetic energy has changed**:
The original kinetic energy was:
\[ KE = \frac{1}{2} m \frac{L^2}{T^2} \]
The new kinetic energy is:
\[ KE' = 4 \cdot KE \]
Therefore, the kinetic energy has increased by a factor of 4.

8. **Consider the effect of doubling both M and L**:
Since we doubled both \( M \) and \( L \), we actually have:
\[ KE' = 2 \cdot 2 \cdot KE = 8 \cdot KE \]

### Final Answer:
The unit of kinetic energy will become **8 times** the original unit.