The units of length, velocity and force are doubled. Which of the following is the correct change in th other units?

Options

A. Unit of time is doubled
B. Unit of mass is doubled
C. Unit of momentum is doubled
D. Unit of energy is doubled

Explanation

To solve the problem, we need to analyze how the doubling of the units of length, velocity, and force affects other physical quantities, specifically mass, time, momentum, and energy.

### Step-by-step Solution:

1. **Understanding the Units**:
- Length (L) is represented as [L].
- Velocity (V) is represented as [L][T]⁻¹ (length per time).
- Force (F) is represented as [M][L][T]⁻² (mass times acceleration).

2. **Doubling the Units**:
- If the unit of length is doubled, we denote this as 2L.
- If the unit of velocity is doubled, we denote this as 2V.
- If the unit of force is doubled, we denote this as 2F.

3. **Finding the Effect on Mass**:
- The formula for force is given by \( F = m \cdot a \), where \( a \) is acceleration.
- Acceleration can be expressed as \( a = \frac{V}{T} \).
- If both length and velocity are doubled, we can analyze the relationship:
- Since \( F \) is doubled and \( V \) is doubled, we can express mass as:
\[ m = \frac{F}{a} = \frac{F}{\frac{V}{T}} = \frac{F \cdot T}{V} \]
- Doubling \( F \) and \( V \) results in:
\[ m' = \frac{2F \cdot T}{2V} = \frac{F \cdot T}{V} = m \]
- Therefore, mass remains unchanged.

4. **Finding the Effect on Time**:
- The time can be derived from the relationship:
\[ T = \frac{L}{V} \]
- If both \( L \) and \( V \) are doubled:
\[ T' = \frac{2L}{2V} = \frac{L}{V} = T \]
- Thus, time also remains unchanged.

5. **Finding the Effect on Momentum**:
- Momentum (P) is given by \( P = m \cdot V \).
- Since mass does not change and velocity is doubled:
\[ P' = m \cdot (2V) = 2(m \cdot V) = 2P \]
- Therefore, momentum is doubled.

6. **Finding the Effect on Energy**:
- Energy (E) can be expressed as work done, which is \( E = F \cdot d \).
- If both force and displacement (length) are doubled:
\[ E' = (2F) \cdot (2L) = 4(F \cdot L) = 4E \]
- Thus, energy is quadrupled.

### Summary of Changes:
- Mass: No change
- Time: No change
- Momentum: Doubled
- Energy: Quadrupled

### Final Answer:
The correct changes in the other units are:
- Mass: No change
- Time: No change
- Momentum: Doubled
- Energy: Quadrupled