A body is thrown vertically upwards which graph represents variation of velocity w.r.t time

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Answer

To solve the problem of determining the graph that represents the variation of velocity with respect to time for a body thrown vertically upwards, we can follow these steps:

### Step 1: Understanding the Motion
When a body is thrown vertically upwards, it initially moves upward with a certain velocity \( u \). As it ascends, it experiences a downward acceleration due to gravity, denoted as \( g \). This acceleration acts in the opposite direction to the motion of the body.

### Step 2: Analyzing Upward Motion
At the highest point of its trajectory, the velocity of the body becomes zero. We can use the equation of motion:
\[ v = u + at \]
where:
- \( v \) is the final velocity (0 at the highest point),
- \( u \) is the initial velocity,
- \( a \) is the acceleration (which is \(-g\) during upward motion),
- \( t \) is the time taken to reach the highest point.

Setting \( v = 0 \):
\[ 0 = u - gt \]
Rearranging gives:
\[ gt = u \quad \Rightarrow \quad t = \frac{u}{g} \]
This means that the time taken to reach the highest point is \( \frac{u}{g} \).

### Step 3: Sketching the Velocity-Time Graph for Upward Motion
- At \( t = 0 \), the velocity is \( u \).
- As time progresses towards \( t = \frac{u}{g} \), the velocity decreases linearly to 0.

This indicates that the graph will start at the point \( (0, u) \) and decrease linearly to the point \( \left(\frac{u}{g}, 0\right) \).

### Step 4: Analyzing Downward Motion
Once the body reaches the highest point, it starts descending. At the highest point, the initial velocity for the downward motion is 0. As it falls, it accelerates downwards due to gravity.

Using the same equation of motion:
\[ v = u + at \]
For the downward motion:
- The initial velocity \( u = 0 \),
- The acceleration \( a = g \).

Thus, the velocity increases as:
\[ v = 0 + gt = gt \]
This means that as time progresses after reaching the highest point, the velocity becomes positive and increases linearly.

### Step 5: Sketching the Velocity-Time Graph for Downward Motion
- At \( t = \frac{u}{g} \), the velocity is 0.
- As time increases, the velocity becomes positive and increases linearly.

### Step 6: Combining Both Motions
The complete graph will consist of:
1. A line segment from \( (0, u) \) to \( \left(\frac{u}{g}, 0\right) \) for the upward motion.
2. A line segment starting from \( \left(\frac{u}{g}, 0\right) \) and increasing positively for the downward motion.

### Conclusion
The graph representing the variation of velocity with respect to time for a body thrown vertically upwards will be a straight line decreasing to zero at the highest point and then a straight line increasing positively as it falls back down.