Projectile motion is a form of motion in which an object or particle is thrown with some initial velocity near the earth’s surface and it moves along a curved path under the action of gravity alone. The path followed by projectile is called its trajectory. An object that is in flight after being thrown is called projectile.

**Example:- (I)** A tennis ball or a baseball in a flight. **(II)** A bullet fired from a rifle etc.

## Assumptions Before The Study Of Projectile Motion

**(I)** There is no frictional resistance of air.

**(II)** The effect due to rotation of Earth and the curvature of the Earth is negligible.

**(III)** The acceleration due to gravity is constant both in magnitude and direction at all points during the motion of projectile.

## Mathematical Analysis Of Projectile Motion

Let an object is projected from point O with velocity u, making an angle θ, with the horizontal direction OX.

While resolving velocity u into two components, we get ucosθ along OX and usinθ along OY. usinθ component try to move the object in Y-direction (upward) while ucosθ component try to move the object in X-direction (Horizontal). So real path of object is curved path.

### Equation Of Path Of A Projectile

Late at any time t, the object reaches at point P(x,y). So

x = Horizontal distance travelled by object in time t.

y = Vertical distance travelled by object in time t.

#### Motion Along Horizontal Direction (OX)

The velocity along OX

#### Motion Along Vertical Direction (OY)

Since, the vertical velocity of the object is decreasing from O to H due to gravity. So, acceleration ay is g.

Therefore, the position of the object at any time t, along the vertical direction OY is-

### Time Of Flight

It is defined as the total time for which projectile is in flight i.e time during the motion of projectile from O to B.

It is denoted by T.

Total time of flight consist of two parts.

**(a)** Time taken by an object from O to H and known as time of ascent (t).

**(b)** Time taken of flight from H to B and known as time of decent (t).

Now, Total time T = t+t= 2t, So t = T/2

Let us consider vertical upward motion of an object from O to H, then we have from equation

### Maximum Height Of A Projectile

It is define as the maximum vertical height attend by an object above the point of projection during its flight.

It is denoted by H

Now, we consider the vertical upward motion of the object from O to H.

### Horizontal Range Of A Projectile

The horizontal range of the projectile is defined as the horizontal distance covered by the projectile during its time of flight.

It is denoted by R.

If object have uniform horizontal velocity u = ucosθ and total time of flight is T, then the horizontal range covered by the object.

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