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.
A car starts from rest with a constant acceleration `a_(1)` for some time and attains a velocity `v_(0)`. Then it immediately retards at a constant rate `a_(2)` to rest. The total time to journey is `t_(0)` Identify the correct velocity-time graph

Options

A.
B.
C.
D. (4). None of these

Answer

To solve the problem, we need to analyze the motion of the car as described in the question. The car undergoes two phases: acceleration and deceleration. Let's break down the steps to identify the correct velocity-time graph.

### Step 1: Understand the Motion Phases
1. **Acceleration Phase**: The car starts from rest and accelerates with a constant acceleration \( a_1 \) until it reaches a maximum velocity \( v_0 \).
2. **Deceleration Phase**: After reaching \( v_0 \), the car immediately starts to decelerate at a constant rate \( a_2 \) until it comes to rest.

### Step 2: Analyze the Velocity-Time Graph
- During the **acceleration phase**, the velocity increases linearly from 0 to \( v_0 \). This indicates that the graph will be a straight line with a positive slope.
- During the **deceleration phase**, the velocity decreases from \( v_0 \) to 0. This also indicates a straight line, but with a negative slope.

### Step 3: Total Time of Journey
- The total time of the journey is \( t_0 \), which is the sum of the time taken to accelerate to \( v_0 \) and the time taken to decelerate to rest.

### Step 4: Identify the Correct Graph
- The correct velocity-time graph should show:
- A straight line from the origin (0,0) to the point (t1, \( v_0 \)), where \( t_1 \) is the time taken to reach maximum velocity.
- A straight line from (t1, \( v_0 \)) to (t0, 0), where \( t_0 \) is the total time taken to come to rest.

### Step 5: Evaluate the Options
1. **Option 1**: A straight line increasing to \( v_0 \) and then a straight line decreasing to 0. This matches our analysis.
2. **Option 2**: A non-linear curve, which indicates changing acceleration. This is incorrect.
3. **Option 3**: A semicircular curve, which also indicates changing acceleration. This is incorrect.
4. **Option 4**: Another non-linear curve, which is incorrect.

### Conclusion
The correct velocity-time graph is **Option 1**, as it accurately represents the motion of the car with constant acceleration followed by constant deceleration.

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