A ball rolls of the top fi a strair way with horizonntal velocity of magnitude ` 1. 8 ms^(-1)`. The steps are ` 0.20 m` high and ` 0.02 m` wide , Which step will the ball it first ? ( g = 10 m//s^@)`.

Explanation

To solve the problem of determining which step the ball will hit first, we can break down the steps as follows:

### Step 1: Understand the Problem
The ball rolls off a staircase with a horizontal velocity of 1.8 m/s. Each step of the staircase has a height of 0.20 m and a width of 0.02 m. We need to find out which step the ball will hit first as it falls.

### Step 2: Analyze Vertical Motion
The vertical motion of the ball can be described using the equation of motion:
\[ h = \frac{1}{2} g t^2 \]
where:
- \( h \) is the height fallen,
- \( g \) is the acceleration due to gravity (10 m/s²),
- \( t \) is the time in seconds.

For the \( n^{th} \) step, the height \( h \) is given by:
\[ h = n \cdot 0.20 \]

### Step 3: Calculate Time to Fall
Rearranging the equation for height gives us:
\[ t = \sqrt{\frac{2h}{g}} \]
Substituting \( h = n \cdot 0.20 \):
\[ t = \sqrt{\frac{2(n \cdot 0.20)}{10}} = \sqrt{\frac{0.04n}{10}} = \sqrt{0.004n} \]

### Step 4: Analyze Horizontal Motion
The horizontal distance traveled by the ball can be described by:
\[ x = u \cdot t \]
where:
- \( x \) is the horizontal distance,
- \( u \) is the horizontal velocity (1.8 m/s).

For the \( n^{th} \) step, the horizontal distance \( x \) is given by:
\[ x = n \cdot 0.02 \]

### Step 5: Set Up the Equation
Now, we can set the two equations equal to each other:
\[ n \cdot 0.02 = 1.8 \cdot t \]
Substituting \( t \) from the vertical motion:
\[ n \cdot 0.02 = 1.8 \cdot \sqrt{0.004n} \]

### Step 6: Solve for \( n \)
Squaring both sides to eliminate the square root:
\[ (n \cdot 0.02)^2 = (1.8)^2 \cdot (0.004n) \]
\[ 0.0004n^2 = 3.24 \cdot 0.004n \]
\[ 0.0004n^2 = 0.01296n \]

Dividing both sides by \( n \) (assuming \( n \neq 0 \)):
\[ 0.0004n = 0.01296 \]
\[ n = \frac{0.01296}{0.0004} = 32.4 \]

### Step 7: Round to the Nearest Whole Number
Since \( n \) must be a whole number (you can't hit a fraction of a step), we round \( 32.4 \) to \( 32 \).

### Conclusion
The ball will hit the **32nd step** first.