Two guns, situated on the top of a hill of height 10 m, fire one shot each with the same speed $$5\sqrt 3 $$ m s^-1 at some interval of time. One gun fires horizontally and other fires upwards at an angle of $$60^\circ $$ with the horizontal. The shots collide in air at a point P. Find

(i) the time interval between the firings, and

(ii) the coordinates of the point P.

Take origin of the coordinate system at the foot of the hill right below the muzzle and trajectories in x-y plane.

The dispalcement x of particle moving in one direction, under the action of a constant force is related to the time t by the equation $$t = \sqrt x + 3$$
where x is in meters and t in seconds. Find

(i) The displacement of the particle when its velocity is zero, and

(ii) The work done by the force in the first 6 seconds.

Answer the following giving reason in brief:

Is the time variation of position, shown in the figure observed in nature?

A car accelerates from rest at a constant rate $$\alpha $$ for some time after which it decelerates at a constant rate $$\beta $$ to come to rest. If the total time lapse is t seconds, evaluate.

(i) maximum velocity reached, and

(ii) the total distance travelled.