1. Constraint Relation

The rope is inextensible, meaning the length $l$ remains constant. Therefore, the velocity components of both the boat and the skier along the direction of the rope must be identical.

• Boat: Velocity is horizontal. The angle with the rope is $\theta$. Component along rope = $v_b \cos\theta$.
• Skier: Velocity is $v_s$. The angle with the rope is $\phi$. Component along rope = $v_s \cos\phi$.

$$ v_b \cos\theta = v_s \cos\phi $$ $$ v_s = v_b \frac{\cos\theta}{\cos\phi} $$

2. Angular Velocity Calculation

The angular velocity $\omega$ of the rope depends on the relative velocity component perpendicular to the rope. This relative velocity causes the rope to rotate.

Components perpendicular to the rope:

• Boat: $v_{b\perp} = v_b \sin\theta$ (Rotates rope down/clockwise)
• Skier: $v_{s\perp} = v_s \sin\phi$ (Rotates rope up/counter-clockwise relative to boat)

The net relative perpendicular velocity is the difference (taking into account the direction that increases the angle):

$$ v_{rel,\perp} = v_s \sin\phi – v_b \sin\theta $$

Angular velocity is linear velocity divided by radius ($l$):

$$ \omega = \frac{v_s \sin\phi – v_b \sin\theta}{l} $$ Substituting $v_s$: $$ \omega = \frac{1}{l} \left( v_b \frac{\cos\theta}{\cos\phi} \sin\phi – v_b \sin\theta \right) $$ $$ \omega = \frac{v_b}{l \cos\phi} ( \sin\phi \cos\theta – \cos\phi \sin\theta ) $$ Using $\sin(A-B) = \sin A \cos B – \cos A \sin B$: $$ \omega = \frac{v_b \sin(\phi – \theta)}{l \cos\phi} $$