A ball rolls off the top of a staircase with a horizontal velocity `u m//s`. If the steps are h meter high and b meter wide, the ball will hit the edge of the nth steps, if:

Options

A. `n = (2hu)/(gb^2)`
B. `n = (2hu^2)/(gb^2)`
C. `n= (2hu^2)/(gb)`
D. `n = (hu)/(gb^2)`

Explanation

To solve the problem of determining when a ball rolling off the top of a staircase will hit the edge of the nth step, we can break it down into the following steps:

### Step 1: Understand the motion of the ball
The ball rolls off the top of the staircase with an initial horizontal velocity \( u \) m/s. The staircase has steps that are \( h \) meters high and \( b \) meters wide. The ball will travel horizontally and vertically as it falls.

### Step 2: Determine the horizontal distance traveled
The horizontal distance the ball travels when it hits the nth step can be expressed as:
\[ x = n \cdot b \]
where \( n \) is the number of steps.

### Step 3: Determine the time taken to reach the nth step
Since the ball moves horizontally with constant velocity \( u \), the time \( t \) taken to travel the horizontal distance \( x \) is given by:
\[ t = \frac{x}{u} = \frac{n \cdot b}{u} \]

### Step 4: Determine the vertical distance fallen
The vertical distance the ball falls when it reaches the nth step is:
\[ y = n \cdot h \]
In vertical motion, the ball starts from rest, so we can use the equation of motion under gravity:
\[ y = \frac{1}{2} g t^2 \]
where \( g \) is the acceleration due to gravity.

### Step 5: Substitute the expression for time into the vertical motion equation
Substituting \( t = \frac{n \cdot b}{u} \) into the vertical motion equation:
\[ n \cdot h = \frac{1}{2} g \left( \frac{n \cdot b}{u} \right)^2 \]

### Step 6: Simplify the equation
Expanding the right side gives:
\[ n \cdot h = \frac{1}{2} g \cdot \frac{n^2 \cdot b^2}{u^2} \]
Rearranging this equation leads to:
\[ 2h \cdot u^2 = g \cdot n \cdot b^2 \]

### Step 7: Solve for \( n \)
Now, we can isolate \( n \):
\[ n = \frac{2h \cdot u^2}{g \cdot b^2} \]

### Conclusion
Thus, the ball will hit the edge of the nth step if:
\[ n = \frac{2h \cdot u^2}{g \cdot b^2} \]

---