Relative Motion

A rigid ring with radius $R$ undergoes pure rolling on a rigid horizontal surface. The center of the ring moves forward horizontally with a constant velocity $v_0$. Consider a point P on the ring that is at the same height as the center.

Particle A moves along the line $y=d$ with a constant velocity $\vec{v}$ of magnitude $v$ parallel to the x-axis. At the instant particle A passes the y-axis, particle B leaves the origin with zero initial speed and a constant acceleration $\vec{a}$ of magnitude $a$. The angle $\theta$ is between the acceleration vector $\vec{a}$ and the positive direction of the y-axis.

A rabbit runs along a straight line at a constant speed of $v_r = 5$ m/s. At a certain moment, a fox spots the rabbit and begins to chase it. The fox maintains a constant speed of $v_f = 4$ m/s, and its velocity vector at any instant points directly towards the rabbit. The distance between them initially decreases and then increases. The minimum distance between the fox and the rabbit is $d_{min} = 30$ m.

Two ships, A and B, leave port at the same time. Ship A travels northwest at 24 knots, and ship B travels at 28 knots in a direction 40° west of south. (1 knot = 1 nautical mile per hour).

During a calm rain, raindrops fall vertically to the ground with a speed of 10 m/s. A cylindrical measuring container with a cross-sectional area of 80 cm² is placed on the ground. After 30 min, the water level of the collected rainwater in the container is 1 cm high. Now, due to wind, the raindrops fall at an angle of $30^{\circ}$ to the vertical.

A river of width $W$ flows due east at a uniform speed $v_w$. A boat with a speed $v_b$ relative to the water leaves the south bank pointed at an angle $\theta$ west of north.

A steamship is sailing due north at a speed of 15 km/h. A person on the ship observes the smoke from the smokestack drifting due east. Later, the ship sails due east at a speed of 24 km/h, and the person observes the smoke drifting due northwest. It is assumed that the wind velocity is constant during both journeys.

As shown in the diagram, a person on a pier of height $h$ above the water pulls a rope attached to a boat at point A. The person pulls the rope, causing it to be reeled in at a constant speed $V$.

A ferry crosses a river that is 600 m wide. The crossing takes 1.0 min and the ferry's resultant path is perpendicular to the river banks. The speed of the ferry in still water is 20 m/s.