To solve the problem step by step, we will analyze the motion of the particle projected upwards and derive the necessary graphs: acceleration-time (a-t), velocity-time (v-t), and displacement-time (s-t).
### Step 1: Determine the time of flight
The particle is projected upwards with an initial velocity \( u = 40 \, \text{m/s} \) and will decelerate due to gravity \( g = 10 \, \text{m/s}^2 \).
The time taken to reach the maximum height can be calculated using the formula:
\[
t_{\text{up}} = \frac{u}{g} = \frac{40 \, \text{m/s}}{10 \, \text{m/s}^2} = 4 \, \text{s}
\]
Since the time taken to go up is equal to the time taken to come down, the total time of flight is:
\[
t_{\text{total}} = t_{\text{up}} + t_{\text{down}} = 4 \, \text{s} + 4 \, \text{s} = 8 \, \text{s}
\]
### Step 2: Calculate the maximum height
The maximum height \( h \) attained by the particle can be calculated using the formula:
\[
h = \frac{u^2}{2g} = \frac{(40 \, \text{m/s})^2}{2 \times 10 \, \text{m/s}^2} = \frac{1600}{20} = 80 \, \text{m}
\]
### Step 3: Plot the acceleration-time graph (a-t graph)
The acceleration of the particle is constant throughout its motion. Since the upward direction is taken as positive and gravity acts downwards, the acceleration is:
\[
a = -g = -10 \, \text{m/s}^2
\]
The a-t graph will be a horizontal line at \( -10 \, \text{m/s}^2 \) from \( t = 0 \) to \( t = 8 \, \text{s} \).
### Step 4: Plot the velocity-time graph (v-t graph)
The velocity of the particle decreases from \( 40 \, \text{m/s} \) to \( 0 \, \text{m/s} \) at \( t = 4 \, \text{s} \) (maximum height) and then increases in the negative direction to \( -40 \, \text{m/s} \) when it strikes the ground at \( t = 8 \, \text{s} \).
The v-t graph will be a straight line starting from \( (0, 40) \) to \( (4, 0) \) and then from \( (4, 0) \) to \( (8, -40) \).
### Step 5: Plot the displacement-time graph (s-t graph)
The displacement starts from \( 0 \, \text{m} \) and increases to \( 80 \, \text{m} \) at \( t = 4 \, \text{s} \) and then decreases back to \( 0 \, \text{m} \) at \( t = 8 \, \text{s} \).
The s-t graph is a concave down parabola that reaches a maximum at \( (4, 80) \) and returns to \( (8, 0) \).
### Summary of Graphs
1. **Acceleration-Time Graph (a-t)**: A horizontal line at \( -10 \, \text{m/s}^2 \) from \( t = 0 \) to \( t = 8 \).
2. **Velocity-Time Graph (v-t)**: A straight line from \( (0, 40) \) to \( (4, 0) \) and then from \( (4, 0) \) to \( (8, -40) \).
3. **Displacement-Time Graph (s-t)**: A concave down parabola reaching a maximum at \( (4, 80) \) and returning to \( (8, 0) \).
