Starting from rest a particle is first accelerated for time `t_1` with constant acceleration `a_1` and then stops in time `t_2` with constant retardation `a_2.` Let `v_1` be the average velocity in this case and `s_1` the total displacement. In the second case it is accelerating for the same time `t_1` with constant acceleration `2a_1` and come to rest with constant retardation `a_2` in time `t_3.` If `v_2` is the average velocity in this case and `s_2` the total displacement, then

Options

A. (a) `v_(2)=2v_(1)`
B. (b) `2v_(1)ltv_(2)lt4v_(1)`
C. (c) `s_(2)=2s_(1)`
D. (d) `2s_(1)lts_(2)lt4s_(1)`

Answer

To solve the problem, we will analyze the motion of the particle in two cases and derive the average velocities and displacements.

### Case 1:
1. **Acceleration Phase**:
- The particle starts from rest and accelerates with constant acceleration \( a_1 \) for time \( t_1 \).
- The final velocity after this phase is given by:
\[ v = a_1 t_1 \]
- The displacement during this phase is:
\[ s_a = \frac{1}{2} a_1 t_1^2 \]

2. **Retardation Phase**:
- The particle then comes to rest with constant retardation \( a_2 \) in time \( t_2 \).
- The relationship between the acceleration and retardation is given by:
\[ a_2 t_2 = a_1 t_1 \implies t_2 = \frac{a_1}{a_2} t_1 \]
- The displacement during this phase is:
\[ s_r = \frac{1}{2} a_2 t_2^2 = \frac{1}{2} a_2 \left(\frac{a_1}{a_2} t_1\right)^2 = \frac{a_1^2 t_1^2}{2 a_2} \]

3. **Total Displacement and Average Velocity**:
- The total displacement \( s_1 \) is:
\[ s_1 = s_a + s_r = \frac{1}{2} a_1 t_1^2 + \frac{a_1^2 t_1^2}{2 a_2} \]
- The average velocity \( v_1 \) is given by:
\[ v_1 = \frac{s_1}{t_1} = \frac{\frac{1}{2} a_1 t_1^2 + \frac{a_1^2 t_1^2}{2 a_2}}{t_1} = \frac{1}{2} a_1 t_1 + \frac{a_1^2 t_1}{2 a_2} \]

### Case 2:
1. **Acceleration Phase**:
- The particle accelerates with constant acceleration \( 2a_1 \) for the same time \( t_1 \).
- The final velocity after this phase is:
\[ v = 2a_1 t_1 \]
- The displacement during this phase is:
\[ s_a = \frac{1}{2} (2a_1) t_1^2 = a_1 t_1^2 \]

2. **Retardation Phase**:
- The particle comes to rest with constant retardation \( a_2 \) in time \( t_3 \).
- The relationship for time \( t_3 \) is:
\[ a_2 t_3 = 2a_1 t_1 \implies t_3 = \frac{2a_1}{a_2} t_1 \]
- The displacement during this phase is:
\[ s_r = \frac{1}{2} a_2 t_3^2 = \frac{1}{2} a_2 \left(\frac{2a_1}{a_2} t_1\right)^2 = \frac{2a_1^2 t_1^2}{a_2} \]

3. **Total Displacement and Average Velocity**:
- The total displacement \( s_2 \) is:
\[ s_2 = s_a + s_r = a_1 t_1^2 + \frac{2a_1^2 t_1^2}{a_2} \]
- The average velocity \( v_2 \) is given by:
\[ v_2 = \frac{s_2}{t_1 + t_3} = \frac{a_1 t_1^2 + \frac{2a_1^2 t_1^2}{a_2}}{t_1 + \frac{2a_1}{a_2} t_1} = \frac{a_1 t_1^2 + \frac{2a_1^2 t_1^2}{a_2}}{t_1 \left(1 + \frac{2a_1}{a_2}\right)} \]

### Conclusion:
- From the calculations, we can derive the relationships:
- \( v_2 = 2 v_1 \)
- \( s_2 = 2 s_1 \)