NEHRU NAGAR

Prepared by:-

Siddhant-[XI-A] Saurabh-[X-C]

The motion of a projectile may be thought of as the result of horizontal and vertical components.

Both the components act independently

What is Projectile Motion?

Projectile Given Angular Projection

Equation of path of projectile Suppose at any time t, the object is at point P (x, y).For motion along horizontal direction, the acceleration ax is

zero. The position of the object at any time t is given by,

Here, x0 = 0, ux = u cos θ, ax = 0

[ Velocity of an object in the horizontal direction is constant]Putting these values in equation (i),⇒ x = ut cos θ

For motion along vertical direction, the acceleration ay is −g.

The position of the object at any time t along the vertical direction is given by,

Here,

∴

Putting the value of t from equation (ii),

⇒

This is an equation of a parabola. Hence, the path of the projectile is a parabola

Time of flight

Total time for which the object is in flight

It is denoted by T.Total time of flight = Time of ascent + Time of descent

∴ T = t + t = 2t [Time of ascent = Time of descent = t]

At the highest point H, the vertical component of velocity becomes zero. For vertical motion of the object (from 0 to H),

⇒

∴

Maximum height •Maximum height ‘h’ reached by the projectileFor vertical upward motion from 0 to H,

Using the relation

we obtain

⇒

That is,

Horizontal Range

•Horizontal distance covered by the object between its point of projection and the point of hitting the ground. It is denoted by R.

‘R’ is the distance travelled during time of flight T.

⇒

⇒

For the maximum horizontal range,sin 2θ = 1 = sin 90°

⇒ 2θ = 90° ⇒ θ = 45° ∴ Maximum horizontal range (Rm) is

Examples of Projectile Motion• Launching a Cannon ball

x

y

0

Frame of reference:

h

v0

Equations of motion:

X

Uniform m.

Y

Accel. m.

ACCL. ax = 0 ay = g = -9.81 m/s2

VELC. vx = v0 vy = g t

DSPL. x = v0 t y = h + ½ g t2

g

Important points :In case of projectile motion , the

horizontal component of a velocity (u cos θ) , acceleration (g) and mechanical energy remains constant while, speed ,velocity , vertical component of velocity (v sin θ) ,momentum ,kinetic energy and potential energy all change.

Velocity and K.E are maximum at the point of projection while minimum (but not zero) at the highest point.

Vector diagrams for projectile motion

TIME HORIZONTAL VELOCITY VERTICAL VELOCITY

0 s 73.1 m/s, right 19.6 m/s, up1 s 73.1 m/s, right 9.8 m/s, up2 s 73.1 m/s, right 0 m/s3 s 73.1 m/s, right 9.8 m/s, down4 s 73.1 m/s, right 19.6 m/s, down5 s 73.1 m/s, right 29.4 m/s, down6 s 73.1 m/s, right 39.2 m/s, down7 s 73.1 m/s, right 49.0 m/s, down

What two factors would affect projectile motion?◦ Angle◦ Initial velocity

Initial Velocity

Angle

Evaluating various infoDetermination of the Time of Flight

Determination of Horizontal Displacementx = vix • t

Determination of the Peak Heighty = viy • t + 0.5 • g • t2

Applications

Applications

The use of projectile motion in sports

Like:- Cricketers know that from which angle

they have to hit to get the maximum range.

For your co-operation