, and that and .

2. A stone is projected at an angle of 30o with velocity 20 m/s. Taking g=10 m/s2 and the axes through O, the point of projection, show that the initial conditions are t=0, x=0, y=0, , . Find the

(a) greatest height reached by the stone;

(b) time of flight and range on the horizontal plane through O.

3. A stone is projected into the air with initial velocity 50 m/s, at an angle , where . Taking g=10 m/s2, find :

(i) the velocity and direction of motion after 1 sec.

(ii) the cartesian equation of the trajectory of the body.

4. A projectile is fired from the top of a cliff of height 65 m; the horizontal and vertical components of its velocity being 80 m/s and 60 m/s, respectively. Taking g=10 m/s2 and the axes through the top of the cliff, find the parametric equations of the path of the projectile after t sec. Also, determine the time of flight and the distance from the foot of the cliff where the projectile strikes the sea at the bottom of the cliff.

5. A tower 70 m high stands on horizontal ground. From its top, a stone is thrown with a speed of 65 m/s at an angle of with the horizontal. Taking g=10 m/s2 , find the

(i) distance from the foot of the cliff at which the stone strikes the ground.

(ii) velocity (in magnitude and direction) at that instant;

(iii) greatest height above the ground level reached by the stone.

6. A particle is thrown horizontally from a tower of height 31.25 m with an initial velocity of 25 m/s. Taking g=10 m/s2 and the axes through the top of the cliff, find the parametric equations of the path of the particle after t sec. and its cartesian equation. Also, find where it will strike the level ground through the foot of the tower and its velocity then, stating the magnitude and direction.

7. A ball is projected horizontally from a cliff of height 245 m , and reached the ground at a horizontal distance of 350 m from the foot of the tower. Determine the initial velocity V, and the velocity (direction and magnitude) on striking the ground (taking g=10 m/s2 ).

8. A bullet is projected with a velocity of 200 m/s at an angle of 30o to the horizontal. Taking g=10 m/s2 , find:

(i) the greatest height attained;

(ii) the range on the horizontal plane and the time of flight;

(iii) the velocity and direction of motion of the bullet when it is at a height of 180 metres.

9. A body is projected from a point O. After t sec, its horizontal and vertical distances from O are given by x = 30t, y = 40t - 5t2 . Find (in m)

(a) the horizontal and vertical components of the initial velocity and its initial speed;

(b) the speed at which it is moving after 1 sec;

(c) when it is travelling horizontally, and the greatest height attained;

(d) the direction in which it is moving when it returns to the horizontal level through O; (how far is it then from O ?)

(e) the cartesian equation of the trajectory of the body.

10. A cricket ball is thrown so that its path is given by x = 20t, y = 5t(3-t) where x and y are measured in metres and t in seconds. Calculate:

(i) Initial velocity and angle of projection;

(ii) the ratio of the greatest height to the range of the ball.

11. At time t, the component velocities of a particle, parallel to the coordinate axes, are given by the equations : and . If the particle is at the origin when t=0, find the values of x and y at time t=2.

12. A stone is projected from the origin O with initial horizontal and vertical velocities , respectively. Write down the equations of motion of the stone, taking the gravity as g . Prove that the time T for the stone to reach the horizontal ground through O is independent of .

13. A golf is hit from a point O with an initial velocity V m/s so that it rises at an angle of 30o to the horizontal. Taking g = 10 m/s2 , show that and .

(a) In its descent, it just clears a wall 11.25 m high, and takes 4.5 sec from the moment of projection to clear the wall. Calculate

(i) value of V

(ii) greatest height reached by the ball;

(b) If the initial velocity of the ball is 50 m/s and the ball were to land on a horizontal green 20 m above the level of the point of impact, calculate the

(i) time of flight

(ii) horizontal distance of the point of impact from the point where the ball lands.

(c) If the golf ball were hit (with an initial velocity of 50 m/s from the top of a hill 45 m high, horizontally across a lake of width 120 m, determine the whether the ball would clear the lake and if so by how much.