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 find the height of the tower from which a stone is dropped, given that it travels 24.5 m in the last second of its journey, we can follow these steps:

### Step 1: Understand the problem
We know that the stone is dropped from the top of a tower, and we need to find the height of the tower (H). The stone travels 24.5 m during the last second of its fall.

### Step 2: Use the equations of motion
We can use the second equation of motion to express the height of the tower. The equation is:
\[ H = \frac{1}{2} g t^2 \]
where:
- \( H \) is the height of the tower,
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( t \) is the total time taken to reach the ground.

### Step 3: Determine the distance traveled in the last second
The distance traveled during the last second can be expressed as:
\[ \text{Distance in last second} = H - H_{t-1} \]
where \( H_{t-1} \) is the height of the stone after \( t-1 \) seconds.

Using the equation of motion for the height after \( t-1 \) seconds:
\[ H_{t-1} = \frac{1}{2} g (t-1)^2 \]

Thus, we can write:
\[ 24.5 = H - H_{t-1} \]
Substituting for \( H_{t-1} \):
\[ 24.5 = H - \frac{1}{2} g (t-1)^2 \]

### Step 4: Substitute \( H \) in terms of \( t \)
From the first equation, we have:
\[ H = \frac{1}{2} g t^2 \]
Substituting this into the equation for the last second:
\[ 24.5 = \frac{1}{2} g t^2 - \frac{1}{2} g (t-1)^2 \]

### Step 5: Simplify the equation
Expanding \( (t-1)^2 \):
\[ (t-1)^2 = t^2 - 2t + 1 \]
Thus:
\[ 24.5 = \frac{1}{2} g \left( t^2 - (t^2 - 2t + 1) \right) \]
\[ 24.5 = \frac{1}{2} g (2t - 1) \]

### Step 6: Solve for \( t \)
Rearranging gives:
\[ 24.5 = g (t - \frac{1}{2}) \]
Substituting \( g = 9.8 \, \text{m/s}^2 \):
\[ 24.5 = 9.8 (t - 0.5) \]
\[ t - 0.5 = \frac{24.5}{9.8} \]
\[ t - 0.5 = 2.5 \]
\[ t = 3 \, \text{s} \]

### Step 7: Calculate the height of the tower
Now substituting \( t = 3 \) back into the height equation:
\[ H = \frac{1}{2} g t^2 \]
\[ H = \frac{1}{2} \times 9.8 \times (3)^2 \]
\[ H = \frac{1}{2} \times 9.8 \times 9 \]
\[ H = 44.1 \, \text{m} \]

### Conclusion
The height of the tower is **44.1 meters**.

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