A stone is dropped from the window of a bus moving at `60 kmh^(-1)`. If the window is 1.96 m high, find the distance along the track, which the stone moves before striking the ground.

Answer

To solve the problem of a stone being dropped from a bus moving at 60 km/h from a height of 1.96 m, we will follow these steps:

### Step 1: Convert the speed of the bus from km/h to m/s
The speed of the bus is given as 60 km/h. To convert this to meters per second (m/s), we use the conversion factor:
\[ 1 \text{ km/h} = \frac{5}{18} \text{ m/s} \]
Thus,
\[ 60 \text{ km/h} = 60 \times \frac{5}{18} \text{ m/s} = \frac{300}{18} \text{ m/s} \approx 16.67 \text{ m/s} \]

### Step 2: Calculate the time taken for the stone to fall
We can use the equation of motion for the vertical direction:
\[ s = ut + \frac{1}{2} a t^2 \]
Here,
- \( s = -1.96 \) m (the stone falls downwards),
- \( u = 0 \) m/s (initial vertical velocity),
- \( a = g = 9.8 \) m/s² (acceleration due to gravity).

Substituting these values into the equation:
\[ -1.96 = 0 \cdot t + \frac{1}{2} \cdot 9.8 \cdot t^2 \]
This simplifies to:
\[ -1.96 = 4.9 t^2 \]
Solving for \( t^2 \):
\[ t^2 = \frac{-1.96}{-4.9} = 0.4 \]
Taking the square root gives:
\[ t = \sqrt{0.4} \approx 0.632 \text{ seconds} \]

### Step 3: Calculate the horizontal distance traveled by the stone
The horizontal distance \( d \) traveled by the stone can be calculated using the formula:
\[ d = u_x \cdot t \]
Where \( u_x \) is the horizontal velocity (which is the speed of the bus) and \( t \) is the time calculated in the previous step:
\[ d = 16.67 \text{ m/s} \cdot 0.632 \text{ s} \]
Calculating this gives:
\[ d \approx 10.52 \text{ m} \]

### Final Answer
The distance along the track that the stone moves before striking the ground is approximately **10.52 meters**.

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