PROJECTILE MOTION

PROJECTILE

A projectile is an object upon which the A projectile is an object upon which the only force acting is gravity. There are a only force acting is gravity. There are a variety of examples of projectiles. An variety of examples of projectiles. An object dropped from rest is a projectile object dropped from rest is a projectile (provided that the influence of air resistance (provided that the influence of air resistance is negligible). An object which is thrown is negligible). An object which is thrown vertically upward is also a projectile vertically upward is also a projectile (provided that the influence of air resistance (provided that the influence of air resistance is negligible). And an object is which is negligible). And an object is which thrown upward at an angle to the horizontal thrown upward at an angle to the horizontal is also a projectile (provided that the is also a projectile (provided that the influence of air resistance is negligible). A influence of air resistance is negligible). A projectile is any object which once projectile is any object which once projected or dropped continues in motion projected or dropped continues in motion by its own inertia and is influenced only by by its own inertia and is influenced only by the downward force of gravity.the downward force of gravity.

Projectile motionProjectile motion

•Motion in a vertical plane Motion in a vertical plane where the only influence is where the only influence is the constant acceleration due the constant acceleration due to gravity.to gravity.

•In projectile motion, the In projectile motion, the horizontal motion and horizontal motion and vertical motion are vertical motion are independent of each other, independent of each other, i.e. they do not affect each i.e. they do not affect each otherother..

This feature allows us to break the motion into This feature allows us to break the motion into two separate one-dimensional problems: one for two separate one-dimensional problems: one for the horizontal motion; the other for the vertical the horizontal motion; the other for the vertical motion.motion.

We will assume that air resistance has no We will assume that air resistance has no effect.effect.

Horizontal and Vertical MotionHorizontal and Vertical Motion

Position, velocity and acceleration are vectors.Position, velocity and acceleration are vectors. These vectors can be separated into components.These vectors can be separated into components.

• Choose x for horizontalChoose x for horizontal

• Choose y for verticalChoose y for vertical

The equations for constant acceleration can be written The equations for constant acceleration can be written separately in each coordinate.separately in each coordinate.

002

21

002

21

002

21

rtvtar

ytvtay

xtvtax

yy

xx

Horizontal motionHorizontal motion

0

0 0

0

cos

x x x

x

x

v v a t

a

v v

210 0 2

0 0 0cos

x xx x v t a t

x x v t

Vertical motionVertical motion

210 0 2

210 0 0 2

0

0 0

(2 15)

sin (4 22)

(2 11)

sin (4 23)

y y

y y y

y

y y v t a t

y y v t gt

v v a t

v v gt

Vector GravityVector Gravity

The gravitational acceleration is The gravitational acceleration is a vector.a vector.

The magnitude is 9.8 m/sThe magnitude is 9.8 m/s22.. The direction is down.The direction is down.

If there is only constant If there is only constant gravitational acceleration in the gravitational acceleration in the --yy direction, the equations direction, the equations simplify for simplify for xx..

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21

002

21

00

rtvtgr

ytvgty

xtvx

y

x

Bullet being fired from a Gun Bullet being fired from a Gun (In Projectile Motion)(In Projectile Motion)

   Bullet:- The motion of the bullet is a Bullet:- The motion of the bullet is a

combination of motion with a constant combination of motion with a constant velocity (along the x-axis) and motion velocity (along the x-axis) and motion with constant acceleration (along the y-with constant acceleration (along the y-axis).axis).

Projectile is a body thrown with an Projectile is a body thrown with an initial velocity in the vertical plane and initial velocity in the vertical plane and then it moves in two dimensions under then it moves in two dimensions under the action of gravity alone without the action of gravity alone without being propelled by any engine or being propelled by any engine or fuel.Its motion is called projectile fuel.Its motion is called projectile motion.The path of a projectile is called motion.The path of a projectile is called its trajectory.its trajectory.

A packet released from an airplane in flight. A golf ball in flight. A bullet fired from a rifle. A jet of water from a hole near the bottom of a water tank.

Examples :-

Moving TargetMoving Target

A target drops as a projectile is fired. To hit the A target drops as a projectile is fired. To hit the target, the projectile should be aimedtarget, the projectile should be aimed• A) Above the target.A) Above the target.

• B) At the target.B) At the target.

• C) Below the target at the point they should collide.C) Below the target at the point they should collide.

• D) Below the target under the point they should collide.D) Below the target under the point they should collide.

Horizontal SpeedHorizontal Speed

cos/

cos

/

0

0

0

00

x

x

x

x

vv

vv

txv

xtvx

The point of collision measures the distance The point of collision measures the distance

dropped since the ball was fired.dropped since the ball was fired.

The time determines the horizontal velocity The time determines the horizontal velocity of the ball.of the ball.

The angle of the launcher gives the total The angle of the launcher gives the total velocity.velocity.

EXAMPLEEXAMPLE

An object is fired from the ground at 100 meters per second at an angle of 30 degrees with the horizontal

Calculate the horizontal and vertical components of the initial velocityAfter 2.0 seconds, how far has the object traveled in the horizontal direction?How high is the object at this point?

SolutionSolution

Part aPart a

Part bPart b

Part cPart c

mssmtvx

t

xv

x

ix

1740.287

s

ms

mvv

sm

smvv

iiy

iix

5030sin100sin

8730cos100cos

0

0

22

2 0.28.92

10.250

2

1s

smss

mtgtvy iy

Throwing a Stone ( In Projectile motion)

A projectile is an object upon which only A projectile is an object upon which only the force of gravity is acting. For our the force of gravity is acting. For our discussions  we assume that the discussions  we assume that the influence of air resistance can be influence of air resistance can be neglected. neglected.

For example, an object dropped from rest at For example, an object dropped from rest at some height is a projectile as long as it some height is a projectile as long as it is in the air. is in the air.

Generally, the projectile is an object thrown Generally, the projectile is an object thrown into the air: into the air:

vertically upward or downward,vertically upward or downward, horizontally,horizontally, or at any angle to the Earth surface.or at any angle to the Earth surface.

An object is controlled by two independent motions. So an An object is controlled by two independent motions. So an object projected horizontally will reach the ground at the object projected horizontally will reach the ground at the same time as an object dropped vertically. No matter how same time as an object dropped vertically. No matter how large the horizontal velocity is, the downward pull of large the horizontal velocity is, the downward pull of gravity is always the same. gravity is always the same.

A projectile launched horizontally has no initial vertical velocity. Therefore, its vertical motion is like that of a dropped object.

Horizontal (x)

x v x t

Vertical (y)

y y0 v0y t 12gt 2

v y v0y gt

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