Projectiles

Projectile Motion Something is fired, thrown, shot, or

hurled near the earth’s surface. Horizontal velocity is constant. Vertical velocity is accelerated. Air resistance is ignored.

1-Dimensional Projectile Definition: A projectile that moves in a

vertical direction only, subject to acceleration by gravity.

Examples: Drop something off a cliff. Throw something straight up and catch it.

You calculate vertical motion only. The motion has no horizontal

component.

2-Dimensional Projectile Definition: A projectile that moves both

horizontally and vertically, subject to acceleration by gravity in vertical direction.

Examples: Throw a softball to someone else. Fire a cannon horizontally off a cliff. Shoot a monkey with a blowgun.

You calculate vertical and horizontal motion.

3-7 Projectile Motion

The speed in the x-direction is constant; in the y-direction the object moves with constant acceleration g.

This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.

Horizontal Component of Velocity

Horizontal Component of Velocity Is constant Not accelerated Not influence by gravity Follows equation:

x = vixt

Vertical Component of Velocity• Undergoes accelerated motion• Accelerated by gravity (9.8 m/s2

down)• Big 3 Kinematic equations apply:

vy = viy gt

y = yi + viyt 1/2gt2

vy2 = viy

2 2g(y – yi)

Horizontal and Vertical

THREE TYPES OF PROJECTILES

Launch angle Definition: The angle at which a

projectile is launched. The launch angle determines what

the trajectory of the projectile will be.

Launch angles can range from -90o (throwing something straight down) to +90o (throwing something straight up) and everything in between.

Zero Launch Angle Projectiles

Zero Launch angle

A zero launch angle implies a perfectly horizontal launch.

vo

Examples of Projectile Motion

Launching a Cannon ball

General Launch Angle Projectiles

General launch angle

vo

Projectile motion is more complicated when the launch angle is not straight up or down (90o or –90o), or perfectly horizontal (0o).

What we will need?We will need to be given the direction of the two components which we will be asked to find.

In this situation, we will find the horizontal and the vertical components (velocity, acceleration, or displacement for this chapter.)

Speed

Without air resistance, a projectile will reach maximum height in the same time it takes to fall from that height to the ground. The deceleration due to gravity going up is the same as the acceleration due to gravity coming down. The projectile hits the ground with the same speed it had when it was projected upward from the ground.

5.6 Projectiles Launched at an Angle

Without air resistance, the speed lost while the cannonball is going up equals the speed gained while it is coming down. The time to go up equals the time to come down.

5.6 Projectiles Launched at an Angle

think!A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration?

5.6 Projectiles Launched at an Angle

think!A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration?

Answer: Its vertical acceleration is g because the force of gravity is downward. Its horizontal acceleration is zero because no horizontal force acts on it.

5.6 Projectiles Launched at an Angle

think!At what point in its path does a projectile have minimum speed?

Answer: The minimum speed of a projectile occurs at the top of its path. If it is launched vertically, its speed at the top is zero. If it is projected at an angle, the vertical component of velocity is still zero at the top, leaving only the horizontal component.

5.6 Projectiles Launched at an Angle

PROJECTILE MOTION IS PARABOLIC

That means y is a function of x an has the formy = ax - bx2, where a & b are constants for any specific parabolic motion.

RELATIVE VELOCITY--The main reason we need vectors!

Trajectory of a 2-D Projectile

x

y

Definition: The trajectory is the path traveled by any projectile. It is plotted on an x-y graph.

Trajectory of a 2-D Projectile

x

y

Mathematically, the path is defined by a parabola.

Trajectory of a 2-D Projectile

x

y

For a projectile launched over level ground, the symmetry is apparent.

Range of a 2-D Projectile

x

y

Range

Definition: The RANGE of the projectile is how far it travels horizontally.

Maximum height of a projectile

x

y

Range

MaximumHeight

The MAXIMUM HEIGHT of the projectile occurs when it stops moving upward.

Maximum height of a projectile

x

y

Range

MaximumHeight

The vertical velocity component is zero at maximum height.

Maximum height of a projectile

x

y

Range

MaximumHeight

For a projectile launched over level ground, the maximum height occurs halfway through the flight of the projectile.

Acceleration of a projectile

g

g

g

g

g

x

y

Acceleration points down at 9.8 m/s2 for the entire trajectory of all projectiles.

Velocity of a projectile

vo

vf

v

v

v

x

y

Velocity is tangent to the path for the entire trajectory.

Velocity of a projectile

vy

vx

vx

vy

vx

vy

vx

x

y

vx

vy

The velocity can be resolved into components all along its path.

Velocity of a projectile

vy

vx

vx

vy

vx

vy

vx

x

y

vx

vy

Notice how the vertical velocity changes while the horizontal velocity remains constant.

Velocity of a projectile

vy

vx

vx

vy

vx

vy

vx

x

y

vx

vy

Maximum speed is attained at the beginning, and again at the end, of the trajectory if the projectile is launched over level ground.

vo -

vo

Velocity of a projectile

Launch angle is symmetric with landing angle for a projectile launched over level ground.

to = 0

t

Time of flight for a projectile

The projectile spends half its time traveling upward…

Time of flight for a projectile

to = 0

t

2t

… and the other half traveling down.

Things to remember! Velocity is always tangent to the

path. You do not know the magnitude…but it is tangent to the path.

Velocity is zero at the instance you change direction. Changing Vx Vx = 0

Changing Vy Vy = 0

3-8 Solving Problems Involving Projectile MotionConceptual Example 3-9: The wrong strategy.

A boy on a small hill aims his water-balloon slingshot horizontally, straight at a second boy hanging from a tree branch a distance d away. At the instant the water balloon is released, the second boy lets go and falls from the tree, hoping to avoid being hit. Show that he made the wrong move. (He hadn’t studied physics yet.) Ignore air resistance.