A body is projected vertically upward from ground. If we neglect the effect of air, then which one of the following is the best representation of variation of speed (v) with time (t)?

Options

A.
B.
C.
D.

Answer

To determine the variation of speed (v) with time (t) for a body projected vertically upward from the ground, we can analyze the motion step by step.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- When a body is projected upwards, it moves against the gravitational force. The acceleration due to gravity (g) acts downwards, opposing the motion.

2. **Initial Velocity**:
- Let the initial velocity of the body when it is projected upwards be \( v_0 \).

3. **Deceleration Due to Gravity**:
- As the body moves upward, it experiences a constant deceleration of \( g \) (approximately \( 9.81 \, \text{m/s}^2 \)). This means that the speed of the body decreases linearly over time until it reaches its maximum height.

4. **Velocity at Maximum Height**:
- At the maximum height, the velocity of the body becomes zero. This occurs after a certain time \( t_{up} \), which can be calculated using the equation:
\[ v = v_0 - g t \]
- Setting \( v = 0 \) gives:
\[ 0 = v_0 - g t_{up} \implies t_{up} = \frac{v_0}{g} \]

5. **After Reaching Maximum Height**:
- After reaching the maximum height, the body will start falling back down. During this phase, the speed will increase linearly as the body accelerates downwards due to gravity.

6. **Graph Representation**:
- The graph of speed (v) versus time (t) will show a linear decrease in speed as the body rises until it reaches the maximum height (where speed = 0) and then a linear increase in speed as it falls back down.

7. **Selecting the Correct Option**:
- Based on the analysis, the best representation of the variation of speed with time is a graph that shows a linear decrease to zero and then a linear increase. This corresponds to the second option in the provided choices.

### Conclusion:
The correct representation of the variation of speed (v) with time (t) for a body projected vertically upward, neglecting air resistance, is a graph that decreases linearly to zero and then increases linearly.

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