Relative Motion

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.

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.

Boat A travels east at a speed of $v_A = 30$ km/h. Boat B travels due north at a speed of $v_B = 45$ km/h.

A river current flows from west to east at 3.0 m/s. A boat is heading in a direction 30° east of north with a speed of 2.0 m/s relative to the water.

Consider a ferry with a speed of 20 m/s in still water, crossing a 600 m wide river. The speed of the river current is 17.32 m/s (as found in the previous problem).

Car A starts at position $x_{A0}$ with a constant velocity $v_A$. Car B starts at the origin with an initial velocity $v_{B0}$ and a constant negative acceleration $a_B$. Assume $v_{B0} > v_A > 0$ and $x_{A0} > 0$.