Graphs are very useful to represent a physical situation. Various quantities can be easily represented on graphs and other quantites can be determined from the graph. For example the slope of velocity-time graph represents instantaneous acceleration. For a motion with constant acceleration slope of velocity-time graph is constant. If acceleration is changing with time, slope with change and thus velocity-time graph will be a non-linear curve. Further the area of velocity-time graph gives displacement.
For the above situation let `t_(1)` be the time of accelerated motion and `t_(2)` be the time or retarded motion, then the correct relation is

Options

A. `a_(1)t_(2)=a_(2)t_(1)`
B. `a_(1)t_(1)=a_(2)t_(2)`
C. `a_(1)t_(1)^(2)=a_(2)t_(2)^(2)`
D. `a_(1)t_(2)^(2)=a_(2)t_(1)^(2)`

Answer

To solve the problem, we will analyze the motion of an object under two different phases: accelerated motion and retarded motion. We will use the first equation of motion to derive the relationship between the times of these two phases.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( t_1 \) be the time of accelerated motion.
- Let \( t_2 \) be the time of retarded motion.
- Let \( a_1 \) be the acceleration during the accelerated motion.
- Let \( a_2 \) be the retardation (negative acceleration) during the retarded motion.
- Let \( V_0 \) be the maximum velocity reached after the accelerated motion.

2. **Apply the First Equation of Motion for Accelerated Motion**:
- The first equation of motion states:
\[ V = u + at \]
- For the accelerated motion, the initial velocity \( u = 0 \) (starting from rest), so:
\[ V_0 = 0 + a_1 t_1 \]
- This simplifies to:
\[ V_0 = a_1 t_1 \quad \text{(Equation 1)} \]

3. **Apply the First Equation of Motion for Retarded Motion**:
- In the retarded motion, the initial velocity is \( V_0 \) and the final velocity is \( 0 \) (the object comes to rest). Thus, we have:
\[ 0 = V_0 - a_2 t_2 \]
- Rearranging gives:
\[ V_0 = a_2 t_2 \quad \text{(Equation 2)} \]

4. **Equate the Two Equations**:
- Since both equations equal \( V_0 \), we can set them equal to each other:
\[ a_1 t_1 = a_2 t_2 \]

5. **Final Relation**:
- The final relation between the acceleration during the accelerated motion and the retardation during the retarded motion, along with their respective times, is:
\[ a_1 t_1 = a_2 t_2 \]

6. **Conclusion**:
- The correct relation between the times of accelerated motion and retarded motion is given by the equation derived above.