Linear Motion

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 $a$.

A particle moves along an x-axis with constant acceleration $a$. The accompanying figure shows the particle's position $x$ as a function of time $t$. The particle is at position $x_0$ at $t=0$, passes the origin at time $t=T$, and is at position $x_s$ at time $t=2T$. The initial position is related to the final scaled position by $x_0 = -x_s/N$, where $N$ is a positive constant determined by the graph's grid.

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.

Table below lists the lengths between adjacent positions of a freely falling small ball, measured from a strobe photograph. The stroboscope flashes once every $T = 1/30$ s. The table below shows the measured displacement $s$ for each interval and the difference in displacement $\Delta s$ between consecutive intervals.

To measure the depth of a well, a small stone is dropped from the wellhead. The sound of the stone hitting the water is heard 2.5 s after it is dropped. The speed of sound in air is over 300 m/s, so for a well that is not too deep, the time it takes for the sound to travel can be neglected.

An elevator is moving upward with an acceleration of $1.22 \text{ m/s}^2$. At the instant its upward velocity is $2.44 \text{ m/s}$, a bolt detaches and falls from the ceiling. The distance between the elevator's ceiling and floor is $2.74 \text{ m}$.

Two identical small balls, A and B, both with mass $m$, are initially at rest. Ball A is a distance $a$ behind ball B. At the same instant, ball A is given an impulse $I$ and ball B is subjected to a constant force $F$. Both the impulse and the force are directed along the line connecting A to B.

As shown in the figure, a right-angled triangle ABC is situated in a vertical plane, with side BC being horizontal and side AB being vertical. The angle between the hypotenuse AC and the horizontal side BC is $\alpha$. A point mass starts from rest at point A and travels to point C under the influence of gravity, constrained to move along specified paths.

Path 1 involves the particle moving along the sides AB and then BC. The time taken for the segment AB is $t_1$, and the time for the segment BC is $t_2$. Path 2 involves the particle moving along the hypotenuse AC. The time taken is $t_3$.

It is assumed that the particle turns at corner B instantaneously and that the magnitude of its velocity is conserved during the turn. The paths are smooth, so there is no friction.

A car initially travels at a constant speed of $10$ m/s for $2$ s. It then accelerates for $3$ s at a rate of $1 \text{ m/s}^2$. Finally, it continues at a new constant speed for $4$ s.

A car is moving at 5 m/s when the brakes are applied. It undergoes uniform acceleration of 0.4 m/s² in the direction opposite to its initial velocity.