A particle is projected horizontally as shown from the rim of a large hemispherical bowl. The displacement of the particle when it strikes the bowl the first time is $R$. Find the velocity of the particle at that instant and the time taken.

(i) The velocity of the bomber at the time of release of the bomb.

(ii) The maximum height attained by the bomb

(iii) The horizontal distance travelled by the bomb before its strikes the ground

(iv) The velocity (magnitude & direction) of the bomb just when it strikes the ground.

A body falls freely from some altitude $H$.At the moment the first body starts falling another body is thrown from the earth's surface which collides with the first at an altitude $h=\mathrm{H}/2$. The horizontal is $l$. Find the initial velocity and the angle at which it was thrown.

The brakes of a train which is travelling at $30{\mathrm{ms}}^{-1}$ are applied as the train passes point $A$. The brakes produce a constant retardation of magnitude $3\lambda {\mathrm{ms}}^{-2}$ unit the speed of the train is reduced to $10{\mathrm{ms}}^{-1}$. The train travels at this speed for a distance and is then uniformly accelerated at $\lambda {\mathrm{ms}}^{-2}$ until it again reaches a speed of $30{\mathrm{ms}}^{-1}$ as it passes point $B$. The time taken by the in travelling from $A$ to $B$, a distance of $4\mathrm{km}$, is $4\min$. Sketch the speed time graph for this motion and calculate:

(i) The value of $\lambda$

(ii) Distance travelled at $10{\mathrm{ms}}^{-1}$

A ship is moving at a constant speed of $10\mathrm{kmhr}$ in the direction of the unit vector $\hat{i}$. Initially, its position vector, relative to a fixed origin is $10\left(-\hat{i}+\hat{j}\right)$ where $\hat{i} \& \hat{j}$ are perpendicular vectors of length $1\mathrm{km}$. Find its position vector relative to the origin at time $t$ hours later. A second ship is moving with constant speed $u {\mathrm{kmhr}}^{-1}$ parallel to the vector $\hat{i}+2\hat{j}$ and is initially at the origin.

(i) If $u=10\sqrt{5} {\mathrm{kmh}}^{-1}$. Find the minimum distance between the ships and the corresponding value of $t$

(ii) Find the value of $u$ for which the ships are on a collision course and determine the value of $t$ at which the collision would occur if no avoiding action were taken.

A balloon starts ascending from the ground at a constant speed of $25{\mathrm{ms}}^{-1}$. After $5\mathrm{s}$, a bullet is shot vertically upward from the ground.

(i) What should be the minimum speed of the bullet so that it may reach the balloon?

(ii) The bullet is shot at twice the speed calculated in (i). Find the height at which it passes the balloon.