A stone is dropped from the top of a tower and travels 24.5 m in the last second of its journey. The height of the tower is

Options

A. 44.1 m
B. 49 m
C. 78.4 m
D. 72 m

Answer

To solve the problem of finding the height of the tower from which the stone is dropped, we can follow these steps:

### Step 1: Understand the Problem
The stone is dropped from the top of a tower and travels 24.5 m in the last second of its fall. We need to find the total height of the tower.

### Step 2: Define Variables
Let:
- \( h \) = height of the tower (in meters)
- \( t \) = total time taken to reach the ground (in seconds)
- \( g \) = acceleration due to gravity = 9.8 m/s²

### Step 3: Use the Equation of Motion
The distance fallen in \( t \) seconds can be expressed using the equation of motion:
\[ h = ut + \frac{1}{2} g t^2 \]
Since the stone is dropped (initial velocity \( u = 0 \)):
\[ h = \frac{1}{2} g t^2 \quad \text{(Equation 1)} \]

### Step 4: Distance Fallen in the Last Second
The distance fallen in the last second can be calculated using the formula:
\[ \text{Distance in last second} = h - \left( u(t-1) + \frac{1}{2} g (t-1)^2 \right) \]
Again, since \( u = 0 \):
\[ \text{Distance in last second} = h - \frac{1}{2} g (t-1)^2 \]
Given that this distance is 24.5 m:
\[ h - \frac{1}{2} g (t-1)^2 = 24.5 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2
From Equation 1, we can express \( h \):
\[ h = \frac{1}{2} g t^2 \]
Substituting this into Equation 2:
\[ \frac{1}{2} g t^2 - \frac{1}{2} g (t-1)^2 = 24.5 \]

### Step 6: Simplify the Equation
Expanding \( (t-1)^2 \):
\[ (t-1)^2 = t^2 - 2t + 1 \]
Thus:
\[ \frac{1}{2} g t^2 - \frac{1}{2} g (t^2 - 2t + 1) = 24.5 \]
This simplifies to:
\[ \frac{1}{2} g (2t - 1) = 24.5 \]

### Step 7: Solve for \( t \)
Rearranging gives:
\[ g (2t - 1) = 49 \]
Substituting \( g = 9.8 \):
\[ 9.8 (2t - 1) = 49 \]
Dividing both sides by 9.8:
\[ 2t - 1 = 5 \]
Thus:
\[ 2t = 6 \quad \Rightarrow \quad t = 3 \text{ seconds} \]

### Step 8: Calculate the Height \( h \)
Now substitute \( t = 3 \) back into Equation 1:
\[ h = \frac{1}{2} g t^2 = \frac{1}{2} \times 9.8 \times (3^2) = \frac{1}{2} \times 9.8 \times 9 = 44.1 \text{ meters} \]

### Final Answer
The height of the tower is \( \mathbf{44.1 \, m} \).

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