The Motion of Projectiles

When a ball is thrown up vertically, it follows the same path and return to the ground. When the same ball is thrown at an angle to the horizontal the ball does not follow the same path to return to the ground . this type of motion is called projectiles.

A ball is allowed to fall from height h and at the same time another ball is projected or thrown horizontally , both ball will reach the ground at the same time.

Therefore the acceleration of the two balls are the same.

A projectiles can be defines as an object thrown into the air or space and moves by itself under the influence of gravity.

Examples of projectiles are :

1. Football kicked into air.
2. Javelin thrown into air.
3. Discuss and shot-put thrown into air
4. Missiles , fired bullets , stones thrown into air.

Definitions

Time of Flight

The time of flight T of a projectiles is defined as the time it takes to reach the same level from which it was projected.

The Maximum Height H

is defined as the highest vertical distance reached measured from horizontal projection plane.

The range R

is defined as the horizontal distance from the point of projection to the point where the projectile  hits the projection plane again.

Angle of projection ɸ

is the angle which the projectiles makes with the horizontal plane at the point of projection.

Initial velocity U

is the fired velocity of the projectiles at the pointy of projection . it has two components:

Formulas

From Equations of motion

V=U +at

Where V= final velocity

U= Initial velocity

a=g = acceleration due to gravity

t=time taken to reach maximum  height.

At maximum height, V=0

    the vertical component of the initial velocity

a=-g  because the motion is against gravity.

From V=U + at

0=Usinɸ -gt

Usinɸ=gt              

= time from ground to maximum height

The time from ground to maximum height is equal to the time from maximum height to the ground

Total time of Flight

MAXIMUM HEIGHT

From the second Equation of motion

V=0 at maximum height.

    the vertical component of the initial velocity

S=H

a=-g

Therefore ,       

=

Where

    the vertical component of the initial velocity

Range R

From the third Equation of motion,

Here, S=R

  the horizontal component of the velocity

                       a=0   No horizontal acceleration.

Therefore,

But               

Maximum Range

At maximum range Sin2ɸ=1, Sin90=1,

Therefore ɸ=45

Height at Maximum Range

At maximum range ɸ=45

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