Uptill now we have been studying the motion of a particle along a straight line i.e. motion in one dimension. Now we consider the motion of a ball, when it is thrown horizontally from certain height. It is observed that the ball travels forward as well as falls downwards, until it strikes something. Suppose that the ball leaves the hand of the thrower at point a (Fig. 3.16 a) and that its velocity at that instant is completely horizontal.

Let this velocity be vx. According to Newton’s first law of motion, there will be no acceleration in horizontal direction. Unless a horizontally directed force acts on the ball. Ignoring the air friction, only force acting on the ball during flight is the force of gravity. There is no horizontal force acting on it. So its horizontal velocity will remain unchanged and will be vx, until the ball hits something. The horizontal motion of ball is simple. The ball moves with constant horizontal velocity component. Hence horizontal distance x is given by

projectile_motion_equation_01The vertical motion of the ball is also not complicated. It will accelerate downward under the force of gravity and hence a =g. this vertical motion is the same as for a freely falling body. Since initial vertical velocity is zero, hence, vertical distance y, using Eq. 3.7, is given by

projectile_motion_equation_02

It is not necessary that an object should be thrown with some initial velocity in the horizontal direction. A football kicked off by a player; a ball thrown by a cricketer and a missile fired from a launching pad, all projected at some angles with the horizontal, are called projectiles.

Suppose_that the_ball_leaves

Fig. 3.16 (a)

Projectile motion is two dimensional motion under constant acceleration due to gravity.

In such cases, the motion of a projectile can be studied easily by resolving it into horizontal and vertical components which are independent of each other. Suppose that a projectile is fired in a direction angle θ with the horizontal by velocity Vi as shown in Fig. 3.16 (b). let components of velocity Vi along the horizontal and vertical directions Vi cos θ and vi sin θ respectively. The horizontal acceleration is ax = 0 because we have neglected air resistance and no other force is acting along this direction whereas vertical acceleration a y = g. hence , the horizontal component Vix remains constant and at any time t, we have

projectile_motion_equation_03

horizontal_by_velocity_Vi

Fig. 3.16 (b)

Now we consider the vertical motion. The initial vertical component of the velocity is sin θ in the upward direction. Using Eq. 3.5 the vertical component of the velocity at any instant t is given by

projectile_motion_equation_04

The magnitude of velocity at any instant is

projectile_motion_equation_05

The angle ф which this resultant velocity makes with the horizontal can be found from

projectile_motion_equation_06In projectile motion one may wish to determine the height to which the projectile rises, the time of flight and horizontal range. These are described below.

Intrusting Information

A photograph of_two_ballsA photograph of two balls released simultaneously from a mechanism that allows one ball to drop freely while the other is projected horizontally, at any time the two balls are at the same level, i.e., their vertical displacements are equal.

Height of the Projectile

In order to determine the maximum height the projectile

projectile_motion_equation_07

Time of flight

The time taken by the body to cove the distance from the place of its projection to the place where it hits the ground at the same level is called the time of flight.

This can be obtained by taking S = h = 0, because the body goes up and comes back to same level, thus covering no vertical distance. If the body is projecting with velocity v making angle θ with a horizontal, then its vertical component will be Vi sin θ. Hence the equation is

projectile_motion_equation_08Where t is the time of flight of the projectile when it is projected from the ground as shown in Fig. 3.16 (b).

Range of the projectile

Maximum distance which a projectile covers in the horizontal direction is called the range of the projectile.

To determine the range R of the projectile, we multiply the horizontal component of the velicity of projection with total time taken by the body after leaving the point of projection. Thus

 projectile_motion_equation_09

But, 2 sin θ cos θ = sin2 θ, thus the range of the projectile depends upon the velocity of projection and the angle of projection.

projectile_motion_equation_10For the range R to be maximum, the factor sin2θ should have maximum value which is 1 when  2 θ = 90⁰ or θ = 45⁰

Point to ponder

Water_is_projectedWater is projected from two rubber pipes at the same speed-from one at an angle of 30⁰ and from the other at 60⁰. Why are the ranges equal?

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