Linear Motion

An object starts from rest ($v_0 = 0$) and undergoes uniform acceleration $a$. Let $s_n$ be the total displacement in the first $n$ seconds (i.e., in a total time of $t=n$ seconds). Let $s_N$ (using Roman numerals in the original text) be the displacement during the $N$-th second (i.e., in the time interval from $t=N-1$ to $t=N$).

A train travels along a straight track from rest at station A to station B, a distance $s$ away, where it comes to a stop. The maximum possible acceleration is $a_1$, and the maximum possible magnitude of deceleration is $a_2$.

A stone is dropped from the top of a tower. In the final second of its fall, it travels a distance of 45 m. The acceleration due to gravity is $g = 10 \text{ m/s}^2$.

A train is traveling at a speed of $72$ km/h. As it approaches a station, it undergoes uniform deceleration and comes to a stop at the platform in $40$ s.

An object is launched vertically, reaching a maximum height $H$. We are interested in the time it spends in a segment of height $h$ (where $h < H$) at the top and bottom of its trajectory.

An object is thrown vertically upwards. It is in view as it passes a window of height $h$ for a total time $T$.

A ball is shot vertically upward from the surface of a planet. A plot of its height $y$ versus time $t$ shows that it reaches a maximum height $y_{max}$ and has a total time of flight $T$ before returning to its starting height.

A small ball is dropped from rest from a height of 19.6 m.

An object is in free fall starting from rest.

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.