A particle starts from rest with constant acceleration. The ratio of space-average velocity to the time average velocity is :-

Options

A. `1//2`
B. `3//4`
C. `4//3`
D. `3//2`

Explanation

To solve the problem of finding the ratio of space-average velocity to time-average velocity for a particle starting from rest with constant acceleration, we can follow these steps:

### Step 1: Define the Variables
- Let the initial velocity \( u = 0 \) (since the particle starts from rest).
- Let the constant acceleration be \( a \).
- Let the time duration be \( t \).

### Step 2: Calculate the Final Velocity
Using the equation of motion:
\[ v = u + at \]
Since \( u = 0 \):
\[ v = at \]

### Step 3: Calculate the Time-Average Velocity
The time-average velocity \( v_{avg, t} \) is defined as the total displacement divided by the total time. The displacement \( s \) can be calculated using:
\[ s = ut + \frac{1}{2} a t^2 \]
Since \( u = 0 \):
\[ s = \frac{1}{2} a t^2 \]
Now, the time-average velocity is:
\[ v_{avg, t} = \frac{s}{t} = \frac{\frac{1}{2} a t^2}{t} = \frac{1}{2} a t \]

### Step 4: Calculate the Space-Average Velocity
The space-average velocity \( v_{avg, s} \) is defined as the total displacement divided by the total distance. The space-average velocity can also be expressed in terms of the final velocity:
\[ v_{avg, s} = \frac{2s}{t} \]
Substituting \( s = \frac{1}{2} a t^2 \):
\[ v_{avg, s} = \frac{2 \cdot \frac{1}{2} a t^2}{t} = \frac{a t^2}{t} = a t \]

### Step 5: Find the Ratio of Space-Average Velocity to Time-Average Velocity
Now, we can find the ratio:
\[ \text{Ratio} = \frac{v_{avg, s}}{v_{avg, t}} = \frac{a t}{\frac{1}{2} a t} = \frac{a t}{\frac{1}{2} a t} = 2 \]

### Step 6: Final Calculation
The ratio of space-average velocity to time-average velocity is:
\[ \text{Ratio} = 2 \]

### Conclusion
Thus, the ratio of space-average velocity to time-average velocity is \( 2 \).

---