To solve the problem of a particle projected vertically upwards with an initial velocity of \(40 \, \text{m/s}\) and to find the displacement and distance covered by the particle in \(6 \, \text{s}\), we can follow these steps:
### Step 1: Determine the time taken to reach the maximum height
The time taken to reach the maximum height (where the velocity becomes zero) can be calculated using the formula:
\[
t = \frac{u}{g}
\]
where:
- \(u = 40 \, \text{m/s}\) (initial velocity)
- \(g = 10 \, \text{m/s}^2\) (acceleration due to gravity)
Substituting the values:
\[
t = \frac{40}{10} = 4 \, \text{s}
\]
### Step 2: Calculate the maximum height reached
The maximum height (displacement upwards) can be calculated using the formula:
\[
h = \frac{u^2}{2g}
\]
Substituting the values:
\[
h = \frac{40^2}{2 \times 10} = \frac{1600}{20} = 80 \, \text{m}
\]
### Step 3: Determine the time taken to fall back down
Since the total time of flight is \(6 \, \text{s}\) and it takes \(4 \, \text{s}\) to reach the maximum height, the time taken to fall back down is:
\[
t_{\text{down}} = 6 - 4 = 2 \, \text{s}
\]
### Step 4: Calculate the distance covered during the downward motion
During the downward motion, the initial velocity at the start of the fall is \(0 \, \text{m/s}\) (at the maximum height). The distance fallen in \(2 \, \text{s}\) can be calculated using:
\[
s = ut + \frac{1}{2}gt^2
\]
Substituting the values:
\[
s = 0 \times 2 + \frac{1}{2} \times 10 \times (2^2) = 0 + \frac{1}{2} \times 10 \times 4 = 20 \, \text{m}
\]
### Step 5: Calculate the total distance covered
The total distance covered by the particle is the sum of the distance going up and the distance coming down:
\[
\text{Total Distance} = \text{Distance Up} + \text{Distance Down} = 80 \, \text{m} + 20 \, \text{m} = 100 \, \text{m}
\]
### Step 6: Calculate the displacement
The displacement is the difference between the final position and the initial position. Since the particle goes up to \(80 \, \text{m}\) and then comes back down \(20 \, \text{m}\), the final position is:
\[
\text{Final Position} = 80 \, \text{m} - 20 \, \text{m} = 60 \, \text{m}
\]
Thus, the displacement is:
\[
\text{Displacement} = 60 \, \text{m}
\]
### Final Answers:
- **Distance covered**: \(100 \, \text{m}\)
- **Displacement**: \(60 \, \text{m}\)
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