A small ball is pushed from a height h along a smooth hemispherical bowl of radius R. With what speed should the ball be pushed so that it just reaches the top of the opposite end of the bowl?

Options

A. `sqrt(2gh)`
B. `sqrt(2g(R+h))`
C. `sqrt(2g(R-h))`
D. None of these

Explanation

To solve the problem of determining the speed at which a small ball should be pushed from a height \( h \) along a smooth hemispherical bowl of radius \( R \) so that it just reaches the top of the opposite end of the bowl, we can use the principle of conservation of mechanical energy.

### Step-by-step Solution:

1. **Identify the Initial and Final States:**
- **Initial State (Point A):** The ball is pushed from a height \( h \) with an initial speed \( u \).
- **Final State (Point B):** The ball reaches the top of the hemisphere, where its speed is \( 0 \).

2. **Write the Energy Conservation Equation:**
The total mechanical energy at point A must equal the total mechanical energy at point B, since there are no non-conservative forces (like friction) acting on the ball.
\[ \text{Potential Energy (PE)} + \text{Kinetic Energy (KE)} \text{ at A} = \text{PE} + \text{KE} \text{ at B} \]
\[ U_A + K_A = U_B + K_B \]

3. **Calculate the Energies:**
- At point A:
- Potential Energy \( U_A = mgh \) (where \( m \) is the mass of the ball)
- Kinetic Energy \( K_A = \frac{1}{2} mu^2 \)
- At point B:
- Potential Energy \( U_B = mgR \) (the height at point B is \( R \))
- Kinetic Energy \( K_B = 0 \) (since the speed at the top is \( 0 \))

4. **Set Up the Equation:**
Substituting the energies into the conservation equation:
\[ mgh + \frac{1}{2} mu^2 = mgR + 0 \]

5. **Simplify the Equation:**
We can cancel the mass \( m \) from all terms (assuming \( m \neq 0 \)):
\[ gh + \frac{1}{2} u^2 = gR \]

6. **Rearranging the Equation:**
Rearranging gives:
\[ \frac{1}{2} u^2 = gR - gh \]
\[ u^2 = 2g(R - h) \]

7. **Solve for \( u \):**
Taking the square root of both sides:
\[ u = \sqrt{2g(R - h)} \]

### Final Answer:
The speed \( u \) at which the ball should be pushed is:
\[ u = \sqrt{2g(R - h)} \]