Relative Motion

An airplane flies a round trip from point A north to point B and then back to A. The airplane's speed relative to the air is $v$. The round-trip time in still air (when air speed relative to ground, $u$, is 0) is $t_0$. Now, consider a wind blowing from south to north with speed $u$ (where $u eq 0$).

A high-speed train travels at initial velocity $v_H$. It is a distance $D$ behind a locomotive moving in the same direction at a constant velocity $v_L$, where $v_H > v_L$. The train's engineer immediately applies the brakes, causing a constant deceleration of magnitude $a$.

Two trains are moving along the same track and are headed toward each other. Their conductors simultaneously apply the brakes. The velocity-time graph in Figure shows their velocities $v$ as a function of time $t$ during the slowing process. The vertical scaling of the graph is set by $v_s = 40.0$ m/s. The braking process begins when the trains are 200 m apart.

In a plane, two straight lines, AB and CD, intersect at an angle $\phi$. The line AB moves with a velocity $v_1$ in a direction perpendicular to itself. The line CD moves with a velocity $v_2$ in a direction perpendicular to itself. The intersection of the two lines is point P.

The lakeshore MN is a straight line. A small boat starts from point A on the shore and travels into the lake at a constant speed, at an angle $\alpha = 15^\circ$ with the shore. At the same time, a person also starts from point A. The person first runs along the shore for some distance and then swims in the water to chase the boat. The person's running speed on the shore is $v_1 = 4$ m/s, and their swimming speed in the water is $v_2 = 2$ m/s.

A train is traveling west on a horizontal track at a speed of 10 m/s. Due to the influence of wind, raindrops are falling with a speed of 5 m/s at an angle of $30^{\circ}$ west of the vertical.

Ship A is located 4.0 km north and 2.5 km east of ship B. Ship A has a velocity of 22 km/h toward the south, and ship B has a velocity of 40 km/h in a direction 37° north of east. Let $t=0$ when the ships are in the positions described.

After falling for a certain time, a parachutist's motion becomes uniform. In still air, a particular parachutist's landing speed is 4.0 m/s. A wind is now blowing, causing the parachutist to move horizontally eastward at a speed of 2.0 m/s.

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.

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