A ball (solid sphere) is thrown down the valley in such a way that it slides with a speed ` v_(0)` initially without rolling. Prove that it will roll without any sliding when its speed falls to `(5/7)v_(0)`. The transition from pure sliding to pure rolling is gradual, so that both sliding and rolling take place during this interval.

Explanation

To solve the problem, we need to analyze the motion of a solid sphere transitioning from sliding to rolling. Here’s a step-by-step solution:

### Step 1: Understand the Initial Conditions
Initially, the solid sphere is sliding down the valley with an initial speed \( v_0 \) and no angular velocity (\( \omega_0 = 0 \)). The friction acting on the sphere is kinetic friction, which opposes the motion.

**Hint:** Identify the forces acting on the sphere and their directions.

### Step 2: Determine the Friction Force
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