A car starts moving on a straight road. It makes the entire journey in three parts. The total time of journey is 25 s. For first part it accelerates at a constant rate `5m//s^(2)`. For second part it moves with a constant speed. For the third part it retards at constant rate `5m//s^(2)` to come to rest.
The correct velocity-time graph is

Options

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B.
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D.

Answer

To solve the problem step by step, we need to analyze the motion of the car in three distinct parts: acceleration, constant speed, and deceleration.

### Step 1: Analyze the first part of the journey (Acceleration)
- The car accelerates at a constant rate of \(5 \, \text{m/s}^2\).
- Let the time taken for this part be \(t_1\).
- The final velocity at the end of this part can be calculated using the formula:
\[ v = u + at \]
where \(u = 0\) (initial velocity), \(a = 5 \, \text{m/s}^2\), and \(t = t_1\).
Thus, the final velocity after the first part is:
\[ v_1 = 0 + 5t_1 = 5t_1 \]

### Step 2: Analyze the second part of the journey (Constant Speed)
- In this part, the car moves with a constant speed \(v_1 = 5t_1\).
- Let the time taken for this part be \(t_2\).
- The distance covered during this part can be calculated as:
\[ d_2 = v_1 \cdot t_2 = (5t_1) \cdot t_2 \]

### Step 3: Analyze the third part of the journey (Deceleration)
- The car decelerates at a constant rate of \(5 \, \text{m/s}^2\) until it comes to rest.
- Let the time taken for this part be \(t_3\).
- The initial velocity for this part is \(v_1 = 5t_1\) and it comes to rest, so the final velocity is \(0\).
- Using the formula:
\[ v = u + at \]
we have:
\[ 0 = 5t_1 - 5t_3 \]
Rearranging gives:
\[ t_3 = t_1 \]

### Step 4: Total time of journey
- The total time of the journey is given as \(25 \, \text{s}\):
\[ t_1 + t_2 + t_3 = 25 \]
Substituting \(t_3 = t_1\):
\[ t_1 + t_2 + t_1 = 25 \implies 2t_1 + t_2 = 25 \]

### Step 5: Determine the relationship between the parts
- From the equations, we have:
\[ t_2 = 25 - 2t_1 \]
- The car accelerates for \(t_1\), moves at constant speed for \(t_2\), and then decelerates for \(t_1\).

### Step 6: Sketch the velocity-time graph
1. **First Part (Acceleration)**: The velocity increases linearly from \(0\) to \(5t_1\) over the time \(t_1\).
2. **Second Part (Constant Speed)**: The velocity remains constant at \(5t_1\) for the duration \(t_2\).
3. **Third Part (Deceleration)**: The velocity decreases linearly from \(5t_1\) to \(0\) over the time \(t_1\).

### Conclusion
The correct velocity-time graph will show:
- A linear increase in velocity during the first part,
- A horizontal line during the second part,
- A linear decrease in velocity during the third part.