A ball rolls down an inclined plane and acquires a velocity `v_(r)` when it reaches the bottom of the plane. If the same ball slides without friction and acquires rolling from the same height down an equally inclined smooth plane and acquires a velocity `v_(s)` (then which of the following statements are not correct?

Options

A. `v_(r)ltv_(s)` because a work is done by the rolling ball against the frictional force.
B. `v_(r)gtv_(s)`, because the angular velocity acquired makes the rolling ball to travel faster
C. `v_(r)=v_(s)` because kinetic energy of the two balls is same at the bottom of the planes.
D. `v_(r)gtv_(s)` because the rolling ball acquires rotational as well as translational kinetic energy.

Explanation

To solve the problem, we need to analyze the motion of a ball rolling down an inclined plane and compare it to the motion of the same ball sliding down a smooth inclined plane. We will derive the velocities \( v_r \) (for rolling) and \( v_s \) (for sliding) and determine which statements about these velocities are correct or incorrect.

### Step-by-Step Solution:

1. **Identify Forces Acting on the Ball:**
- When the ball rolls down the inclined plane, the forces acting on it are:
- Gravitational force \( mg \) acting downward.
- The component of gravitational force along the incline: \( mg \sin \theta \).
- The frictional force \( f \) that allows rolling.
- For the sliding case, the only force acting along the incline is \( mg \sin \theta \) since there is no friction.

2. **Write the Equations of Motion:**
- For the rolling ball:
\[ mg \sin \theta - f = ma_1 \]
- For the sliding ball:
\[ mg \sin \theta = ma_2 \]
- Here, \( a_1 \) is the acceleration of the rolling ball and \( a_2 \) is the acceleration of the sliding ball.

3. **Relate Accelerations:**
- Since the sliding ball experiences no friction, it will have a greater acceleration than the rolling ball:
\[ a_2 > a_1 \]

4. **Use Kinematic Equations to Find Velocities:**
- For the rolling ball:
\[ v_r^2 = u^2 + 2a_1 s \quad (u = 0) \]
\[ v_r^2 = 2a_1 s \]
- For the sliding ball:
\[ v_s^2 = u^2 + 2a_2 s \quad (u = 0) \]
\[ v_s^2 = 2a_2 s \]

5. **Compare the Velocities:**
- Since \( a_2 > a_1 \), we can conclude that:
\[ v_s^2 = 2a_2 s > v_r^2 = 2a_1 s \]
- Therefore, \( v_s > v_r \).

6. **Evaluate the Statements:**
- The question asks which statements are not correct regarding the velocities \( v_r \) and \( v_s \). Based on our analysis:
- \( v_r < v_s \) is correct.
- The statement regarding work done against friction for the rolling ball is also correct, as the work done is less due to the rolling motion.

### Conclusion:
Based on the analysis, the statements that are not correct can be identified as those that contradict the derived relationship \( v_s > v_r \).