To solve the problem of how long it takes for a stone thrown vertically upward from a height of 40 meters to reach the ground, we can break the solution down into several steps:
### Step 1: Understand the motion of the stone
The stone is thrown upward with an initial velocity of \(10 \, \text{m/s}\) from the top of a cliff that is \(40 \, \text{m}\) high. The stone will first rise to a maximum height before falling back down to the ground.
### Step 2: Calculate the time taken to reach maximum height (T1)
At the maximum height, the final velocity (\(v\)) of the stone will be \(0 \, \text{m/s}\). We can use the first equation of motion:
\[
v = u + at
\]
Where:
- \(u = 10 \, \text{m/s}\) (initial velocity)
- \(v = 0 \, \text{m/s}\) (final velocity at maximum height)
- \(a = -g = -10 \, \text{m/s}^2\) (acceleration due to gravity, taken as negative because it acts downward)
Substituting the values:
\[
0 = 10 - 10 \cdot T_1
\]
Rearranging gives:
\[
10 \cdot T_1 = 10 \implies T_1 = 1 \, \text{s}
\]
### Step 3: Calculate the maximum height reached
To find the maximum height reached by the stone, we can use the second equation of motion:
\[
s = ut + \frac{1}{2} a t^2
\]
Where:
- \(s\) is the displacement (maximum height reached above the cliff)
- \(u = 10 \, \text{m/s}\)
- \(a = -10 \, \text{m/s}^2\)
- \(t = T_1 = 1 \, \text{s}\)
Substituting the values:
\[
s = 10 \cdot 1 + \frac{1}{2} \cdot (-10) \cdot (1)^2
\]
\[
s = 10 - 5 = 5 \, \text{m}
\]
### Step 4: Calculate the total height from which the stone falls
The total height from which the stone falls is the height of the cliff plus the maximum height reached:
\[
\text{Total height} = 40 \, \text{m} + 5 \, \text{m} = 45 \, \text{m}
\]
### Step 5: Calculate the time taken to fall to the ground (T2)
Now, we can calculate the time taken to fall from this height using the second equation of motion again:
\[
s = ut + \frac{1}{2} a t^2
\]
Where:
- \(s = 45 \, \text{m}\) (total height)
- \(u = 0 \, \text{m/s}\) (initial velocity when falling)
- \(a = 10 \, \text{m/s}^2\) (acceleration due to gravity)
Substituting the values:
\[
45 = 0 \cdot T_2 + \frac{1}{2} \cdot 10 \cdot T_2^2
\]
This simplifies to:
\[
45 = 5 T_2^2 \implies T_2^2 = \frac{45}{5} = 9 \implies T_2 = 3 \, \text{s}
\]
### Step 6: Calculate the total time taken to reach the ground
The total time taken to reach the ground is the sum of the time taken to reach maximum height and the time taken to fall to the ground:
\[
\text{Total time} = T_1 + T_2 = 1 \, \text{s} + 3 \, \text{s} = 4 \, \text{s}
\]
### Final Answer
The total time taken by the stone to reach the ground is **4 seconds**.
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