Relative Motion

As shown in the figure, a semi-cylinder with radius R undergoes uniformly accelerated motion with a constant horizontal acceleration $a$. A vertical rod, constrained to move only in the vertical direction, rests on the curved surface of the semi-cylinder. At the instant when the semi-cylinder's horizontal velocity is $v$, the contact point P between the rod and the semi-cylinder is at an angular position $\theta$ with respect to the vertical axis.

As shown in Figure, two small rings, O and O', are on fixed vertical rods AB and A'B', respectively. An inextensible string passes through ring O', with its ends tied to a fixed point A' and ring O. Ring O' moves downwards with a constant speed $v$.

A spool is formed by a small cylinder of radius $r$ coaxially fixed to a large cylinder of radius $R$. A string is wound on the inner cylinder. The string is drawn out, passes over a pulley Q, and its end is pulled with a constant velocity $v$. Simultaneously, the spool rolls without slipping on a horizontal surface. The segment of the string from the spool to the pulley, PQ, makes an angle $\varphi$ with the horizontal, as shown in the diagram.

As shown in the figure, two horizontal tracks in the same vertical plane are separated by a distance $h$. On the tracks are two objects A and B, connected by a light, inextensible string passing over a fixed pulley O. Object A moves on the lower track at a constant speed $v$. At the instant the string segment BO makes an angle of $30^\circ$ with the horizontal, a small water droplet P, initially at the midpoint of BO, detaches from the string.

An airplane is in uniform linear motion in the horizontal direction. An object is dropped from the plane. Air resistance is negligible.

In an overhead view, two jeeps, P and B, race past a stationary observer A. Let East be the positive x-direction and North be the positive y-direction. Relative to observer A, jeep B travels at a constant speed $v_B$ at an angle $\theta_2$ south of east. Relative to observer A, jeep P accelerates from rest at a constant rate $a_P$ at an angle $\theta_1$ north of east. At a certain time, jeep P has a speed $v_P$.

A person is riding a bicycle north at a speed of 6 m/s. The cyclist feels an apparent wind with a speed of 8 m/s, directed to the east.

As shown in the figure, one end of a thin rope is fixed at point A. A weight B is attached to the rope at a distance $a$ from A, making the length of segment AB constant, $l_{AB} = a$. The other end of the rope passes over a fixed pulley at point C. Points A and C lie on the same horizontal line. The free end of the rope is pulled with a constant speed $v$. At the instant shown, the rope segments AB and BC make angles $\alpha$ and $\beta$ with the horizontal, respectively.

A train is moving east at a speed of 8 m/s. Raindrops are falling vertically downwards with a speed of 6 m/s relative to the ground.

An airplane's speed in still air is 180 km/h. The wind speed is 10 m/s.