A staircase has 5 steps each 10 cm high and 10 cm wide. What is the minimum horizontal velocity to be given to the ball so that it hits directly the lowest plane from the top of the staircase? `(g= 10ms^(-2))`

Options

A. `2 ms^(-1)`
B. `1 ms^(-1)`
C. `sqrt(2) ms^(-1)`
D. `(1)/(2) ms^(-1)`

Answer

To solve the problem of finding the minimum horizontal velocity required for a ball to hit the lowest plane from the top of a staircase with 5 steps, each 10 cm high and 10 cm wide, we can follow these steps:

### Step 1: Determine the Total Height
The total height of the staircase can be calculated by multiplying the height of each step by the number of steps.

\[ \text{Total Height} = \text{Height of each step} \times \text{Number of steps} = 10 \, \text{cm} \times 5 = 50 \, \text{cm} \]

Convert this height into meters:

\[ \text{Total Height} = 50 \, \text{cm} = 0.50 \, \text{m} \]

### Step 2: Calculate the Time of Flight
Using the equation of motion for vertical displacement, we can find the time it takes for the ball to fall this height. The formula is:

\[ h = \frac{1}{2} g t^2 \]

Where:
- \( h = 0.50 \, \text{m} \)
- \( g = 10 \, \text{m/s}^2 \)

Rearranging the formula to solve for \( t \):

\[ 0.50 = \frac{1}{2} \times 10 \times t^2 \]

\[ 0.50 = 5 t^2 \]

\[ t^2 = \frac{0.50}{5} = 0.10 \]

\[ t = \sqrt{0.10} = 0.316 \, \text{s} \]

### Step 3: Determine the Horizontal Distance
The horizontal distance that the ball needs to travel is equal to the total width of the staircase, which can also be calculated:

\[ \text{Total Width} = \text{Width of each step} \times \text{Number of steps} = 10 \, \text{cm} \times 5 = 50 \, \text{cm} = 0.50 \, \text{m} \]

### Step 4: Calculate the Minimum Horizontal Velocity
Using the formula for horizontal motion, where horizontal distance \( d \) is given by:

\[ d = u \cdot t \]

Where:
- \( d = 0.50 \, \text{m} \)
- \( t = 0.316 \, \text{s} \)

Rearranging to solve for \( u \):

\[ u = \frac{d}{t} = \frac{0.50}{0.316} \approx 1.58 \, \text{m/s} \]

### Conclusion
The minimum horizontal velocity \( u \) required for the ball to hit the lowest plane from the top of the staircase is approximately:

\[ \boxed{1.58 \, \text{m/s}} \]