Vertical Motion Under Gravity

A stone falls freely from rest and the total distance covered by it in the last second of its motion equals the distance covered by it in the first three seconds of its motion. The stone remains in air for (Use $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

Three particles are projected simultaneously and in the same vertical plane with different velocities and at different angles. Show that the area of the triangle formed by the particles at time $t$ is proportional to $t^2$.

A ball of mass $100 \mathrm{g}$ is projected vertically upward from the ground with a velocity of $49 {\mathrm{ms}}^{-1}$. At the same time another identical ball is dropped from a height of $98 \mathrm{m}$ to fall freely along the same path as that followed by the first ball. After some time, the two balls collide and stick together and finally fall to the ground. Find the time of fall of the masses.

A stone is let fall from the top of a cliff. It describes last $h$ height in time $t$ second. Show that the total time of fall of the stone is $\left(\frac{t}{2}+\frac{h}{gt}\right)s$.

Two particles $A$ and $B$ having different masses are projected from a tower with same speed. $A$ is projected vertically upward and $B$ downward. On reaching the ground

From a tower of height $H$, a particle is thrown vertically upwards with a speed $u$. The time taken by the particle, to hit the ground, is $n$ times that taken by it to reach the highest point of its path. The relation between $H, u$ and $n$ is

From the top of a tower, $80 \mathrm{m}$ high from the ground, a stone is thrown in the horizontal direction with a velocity $8{ \mathrm{ms}}^{-1}$. The stone reaches the ground after a time $t$ and falls at a distance of $\text{'}d\text{'}$ from the foot of the tower. Assuming $\mathrm{g}=10{ \mathrm{ms}}^{-2}$, the time $t$ and distance $d$ are given respectively by

A stone falling from the top of a vertical tower has descended $xm$ when another is let fall from a point $y m$ below the top. If they fall from rest and reach the ground together, show that the height of the tower is

A stone is dropped from the top of a tower of height $h\mathit{.}\mathit{ }n$ seconds later, a second stone is thrown downwards from the same point with a vertically downward velocity $u$. Show that the two stones will reach the foot of the tower simultaneously if $8h{\left(u-gn\right)}^2=g{n}^2{\left(2u-gn\right)}^2$.

A particle is projected vertically upward. Prove that it will be at $\frac{5}{9}$ of its greatest height at times which are in the ratio $1:3$.

Show that the height fallen through by a body in time $t$ is $n^2$ times the height fallen through by the same in time $t/n$.

A ball is thrown vertically upward. Ignoring the air resistance, which one of the following plots represents the velocity-time plot, Fig. for the period ball remains in air?

The water drops fall at regular intervals from a tap $5 \mathrm{m}$ above the ground. The third drop is leaving the top at the instant the first drop touches the ground. How far above the ground is the second drop at that instant 

A body falling freely from rest describes distances $S_1,{ S}_2$ and $S_3$ in the first, second and third seconds of its fall. The ratio of $S_1:S_2:S_3$ is
 

A body is thrown vertically upward. The two instants when the body will reach $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.th$ of the greatest height, are in the ratio of

When a ball is thrown vertically upward from the top point $A$ of a tower, it takes $t_1$ second to hit the ground. If it is thrown downward from $A$ with same velocity it takes $t_2$ second to hit the ground. If the stone is allowed to fall freely from $A$, the time it will take to reach the ground is

From the top of a tower, a stone is let fall. During last $1$ second of its fall the stone clears $\frac{5}{9}th$ of the height of the tower. The time of fall of the stone is