PROJECTILE MOTION

Sonia

Content

Definition of Projectile

Definition of Projectile Motion

Types of Projectile Motion

Examples of Projectile motion

Derivation of projectile motion in 2-D

Factors Affecting Projectile Motion

Definition Of Projectile

A projectile is any object that once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

By definition, a projectile has a single force that acts upon it - the force of gravity. If there were any other force acting upon an object, then that object would not be a projectile. Thus, the free-body diagram of a projectile would show a single force acting downwards and labeled force of gravity (or simply Fgrav). Regardless of whether a projectile is moving downwards, upwards, upwards and rightwards, or downwards and leftwards, the free-body diagram of the projectile is still as depicted in the diagram at the right.

Fgrav

Free-body diagram of a projectile

Definition Of Projectile Motion

Projectile motion is a form of motion in which an object or particle (called a projectile) is thrown near the earth's surface, and it moves along a curved path under the action of gravity only. Example: Parabolic water trajectory

Types Of Projectile Motion• Horizontal

– Motion of a ball rolling freely along a level surface

– Horizontal velocity is ALWAYS constant

• Vertical– Motion of a freely falling object– Force due to gravity– Vertical component of velocity

changes with time• Parabolic

– Path traced by an object accelerating only in the vertical direction while moving at constant horizontal velocity

Examples Of Projectile Motion

Launching a Cannon ball

Examples Of Projectile Motion

Object thrown upward from a car moving in a horizontal direction

Derivation Of Projectile Motion in 2-D

The initial velocityIf the projectile is launched with an initial velocity , then it can be written as

The components and can be found if the angle, is known:

If the projectile's range, launch angle, and drop height are known, launch velocity can be found using Newton's formula

The launch angle is usually expressed by the symbol theta, but often the symbol alpha is used.

Initial velocity of parabolic throwing

Components of initial velocity of parabolic throwing

Derivation Of Projectile Motion in 2-D

Kinematic quantities of projectile motionIn projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other.

AccelerationSince there is no acceleration in the horizontal direction, the velocity in the horizontal direction is constant, being equal to

The vertical motion of the projectile is the motion of a particle during its free fall. Here the acceleration is constant, being equal to The components of the acceleration are:

Derivation Of Projectile Motion in 2-D

VelocityThe horizontal component of the velocity of the object remains unchanged throughout the motion. The vertical component of the velocity increases linearly, because the acceleration due to gravity is constant. The accelerations in the and directions can be integrated to solve for the components of velocity at any time , as follows:

The magnitude of the velocity (under the Pythagorean theorem):

Derivation Of Projectile Motion in 2-D

Displacement Displacement and coordinates of parabolic throwingAt any time , the projectile's horizontal and vertical displacement:

The magnitude of the displacement:Displacement and coordinates of parabolic throwing

Derivation Of Projectile Motion in 2-D

Parabolic trajectoryConsider the equations,

If t is eliminated between these two equations the following equation is obtained:

This equation is the equation of a parabola. Since ,and are constants, the above equation is of the form

in which a and b are constants. This is the equation of a parabola, so the path is parabolic. The axis of the parabola is vertical.

Derivation Of Projectile Motion in 2-D

The maximum height of projectileThe highest height which the object will reach is known as the peak of the object's motion. The increase of the height will last, until that is, Time to reach the maximum height:

From the vertical displacement of the maximum height of projectile:

Maximum height of projectile

Derivation Of Projectile Motion in 2-D

The maximum distance of projectileIt is important to note that the Range and the Maximum height of the Projectile does not depend upon mass of the trajected body. Hence Range and Maximum height are equal for all those bodies which are thrown by same velocity and direction. Air resistance does not affect displacement of projectile.

The horizontal range d of the projectile is the horizontal distance the projectile has travelled when it returns to its initial height (y = 0).

Maximum distance of projectile

Derivation Of Projectile Motion in 2-D

Time to reach ground:

From the horizontal displacement the maximum distance of projectile: So

Note that d has its maximum value when

which necessarily corresponds to

Maximum distance of projectile

Factors Affecting Projectile Motion

What two factors would affect projectile motion?

– Angle– Initial velocity

Initial VelocityAngle

Thank You!