In two different system of unit an acceleration is represented by the same number, whilst a velocity is represented by numbers in the ratio `1:3`. The ratio of unit of length and time are

Options

A. `1/3, 1/9`
B. `1/9, 1/3`
C. `1, 1`
D. None of these

Explanation

To solve the problem step by step, we need to analyze the relationship between acceleration, velocity, length, and time in two different systems of units.

### Step 1: Understand the given information
We know that:
- Acceleration is represented by the same number in both systems.
- Velocity is represented by numbers in the ratio of 1:3.

### Step 2: Define the units
Let's denote:
- The first system of units for velocity as \( V_1 \) and for acceleration as \( A_1 \).
- The second system of units for velocity as \( V_2 \) and for acceleration as \( A_2 \).

From the problem, we have:
- \( V_1 : V_2 = 1 : 3 \)
- \( A_1 = A_2 \)

### Step 3: Relate length and time to velocity and acceleration
We know the following relationships:
- Velocity (\( V \)) is defined as \( V = \frac{L}{T} \) (where \( L \) is length and \( T \) is time).
- Acceleration (\( A \)) is defined as \( A = \frac{V}{T} = \frac{L}{T^2} \).

### Step 4: Express Length and Time in terms of Velocity and Acceleration
From the definitions:
1. Length can be expressed as:
\[ L = V \cdot T \]
2. Time can be expressed as:
\[ T = \frac{V}{A} \]

### Step 5: Substitute the ratios into the equations
Using the ratio of velocities:
- Let \( V_1 = 1 \) and \( V_2 = 3 \).

Now, substituting into the equations for length and time:
1. For the first system:
\[ L_1 = V_1 \cdot T_1 = 1 \cdot T_1 = T_1 \]
\[ T_1 = \frac{V_1}{A_1} = \frac{1}{A_1} \]

2. For the second system:
\[ L_2 = V_2 \cdot T_2 = 3 \cdot T_2 \]
\[ T_2 = \frac{V_2}{A_2} = \frac{3}{A_2} \]

### Step 6: Set up the ratios
Since \( A_1 = A_2 \), we can denote \( A = A_1 = A_2 \). Thus:
- For the first system:
\[ T_1 = \frac{1}{A} \]
- For the second system:
\[ T_2 = \frac{3}{A} \]

### Step 7: Find the ratio of Length and Time
Now we can find the ratio of lengths and times:
- The ratio of lengths:
\[ \frac{L_1}{L_2} = \frac{T_1}{3 \cdot T_2} = \frac{\frac{1}{A}}{3 \cdot \frac{3}{A}} = \frac{1}{9} \]
- The ratio of times:
\[ \frac{T_1}{T_2} = \frac{\frac{1}{A}}{\frac{3}{A}} = \frac{1}{3} \]

### Final Result
Thus, the ratio of the units of length and time is:
\[ \frac{L_1}{L_2} : \frac{T_1}{T_2} = \frac{1}{9} : \frac{1}{3} \]

This means the ratio of the unit of length to the unit of time is \( 1:3 \).