Velocity (v) and acceleration (a) in two systems of units 1 and 2 are related as `v_(2)=(n)/(m^(2))v_(1) and a_(2)=(a_(1))/(mn)` respectively. Here m and n are constants. The relations for distance and time in two systems respectively are:

Options

A. `(n^(3))/(m^(3))L_(1)=L_(2) and (n^(2))/(m)T_(1)=T_(2)`
B. `L_(1)=(n^(4))/(m^(2))L_(2) and T_(1)=(n^(2))/(m)T_(2)`
C. `L_(1)=(n^(2))/(m)L_(2) and T_(1)=(n^(4))/(m^(2))T_(2)`
D. `(n^(2))/(m)L_(1)=L_(2) and (n^(4))/(m^(2))T_(1)=T_(2)`

Explanation

To solve this problem, we need to find the relations for distance and time between two systems of units given the relations for velocity and acceleration.

Given:
\[ v_2 = \frac{n}{m^2} v_1 \]
\[ a_2 = \frac{a_1}{mn} \]

Let's find the relations step-by-step.

### Step 1: Finding the Relation for Time

1. **Use the given relation for velocity:**
\[ v_2 = \frac{n}{m^2} v_1 \]

This implies:
\[ \frac{v_2}{v_1} = \frac{n}{m^2} \]

2. **Use the given relation for acceleration:**
\[ a_2 = \frac{a_1}{mn} \]

This implies:
\[ \frac{a_2}{a_1} = \frac{1}{mn} \]

3. **Use the relation between velocity, acceleration, and time:**
\[ v = a \cdot t \]

Therefore:
\[ v_2 = a_2 \cdot t_2 \]
\[ v_1 = a_1 \cdot t_1 \]

4. **Divide the equations:**
\[ \frac{v_2}{v_1} = \frac{a_2 \cdot t_2}{a_1 \cdot t_1} \]

Substitute the known ratios:
\[ \frac{n}{m^2} = \frac{\frac{1}{mn} \cdot t_2}{t_1} \]

5. **Solve for \(\frac{t_2}{t_1}\):**
\[ \frac{n}{m^2} = \frac{t_2}{mn \cdot t_1} \]
\[ t_2 = \frac{n^2}{m} \cdot t_1 \]

### Step 2: Finding the Relation for Distance

1. **Use the relation between velocity, distance, and time:**
\[ v = \frac{d}{t} \]

Therefore:
\[ d = v \cdot t \]

2. **Express distances in both systems:**
\[ d_2 = v_2 \cdot t_2 \]
\[ d_1 = v_1 \cdot t_1 \]

3. **Divide the equations:**
\[ \frac{d_2}{d_1} = \frac{v_2 \cdot t_2}{v_1 \cdot t_1} \]

4. **Substitute the known ratios:**
\[ \frac{d_2}{d_1} = \left(\frac{n}{m^2}\right) \cdot \left(\frac{n^2}{m}\right) \]

5. **Simplify:**
\[ \frac{d_2}{d_1} = \frac{n^3}{m^3} \]
\[ d_2 = \frac{n^3}{m^3} \cdot d_1 \]

### Final Relations:

- **Time:**
\[ t_2 = \frac{n^2}{m} \cdot t_1 \]

- **Distance:**
\[ d_2 = \frac{n^3}{m^3} \cdot d_1 \]