Projectile Motion

What is a PROJECTILE?

• An object that is projected (launched)• It continues in motion due to its own inertia, • Is only acted upon by gravity• No force in the x-direction, only in the y-

direction (gravity)• Gravity ALWAYS acts in the y-direction, and

ONLY the y-direction

So Projectile Motion…

• Describes the motion of an object in TWO dimensions

• We will only consider projectiles that stay close to Earth (so ag = -9.8 m/s2)

• We will continue to disregard air resistance

Some Vocabulary…

• PROJECTILE– The object being launched/thrown/projected

• TRAJECTORY– The path that the projectile follows

• RANGE– The horizontal displacement of the projectile

• MAXIMUM HEIGHT– The vertical displacement of the projectile at the

top of its flight

The Components of Projectile Motion - Velocity

• We ALWAYS break projectile motion down into its x- and y-components– INITIAL VELOCITY: vi

• Use cos and sin to find vix and viy

– vix = vicosθ (initial velocity * cos angle)

– viy = visinθ (initial velocity * sin angle) vi

viy

vix

The Components of Projectile Motion - Velocity

• For a baseball lobbed with an initial velocity of 30 m/s and an angle of 75 degrees, find the initial horizontal and vertical velocities.

– Horzontal: x axis– Use cos θ– vix = vi * cos θ

– vix = 30 * cos 75

– vix = 30 * .26

– vix = 7.76 m/s

- Vertical: y axis- Use sin θ- viy = vi * sin θ- viy = 30 * sin 75- viy = 30 * .97 - viy = 29 m/s

The Components of Projectile Motion -Acceleration

• ACCELERATION: Always:– ay = -9.8 m/s2 (vertical acceleration)

– ax = 0 m/s2 (horizontal acceleration)

–Why? Think gravity and Newton’s 1st Law…

Remember Freefall???

• Recall that…– a = -9.8 m/s2, regardless if the object is moving up

or moving down– The ONLY force acting on the object is GRAVITY

• Projectile Motion has the same conditions, and moves in the x-direction simultaneously.

What does this look like?

• For horizontally projected objects:

• For objects projected at an angle:

VERY IMPORTANT!!!

• The components act INDEPENDENTLY of one another!!!

• When we combine the x- and y-components, we get the characteristic parabola-shape– Vx remains constant (a = 0)

– Vy changes because of gravity (a = -9.8 m/s2)

How do I determine the direction?• If an object is projected at an angle, the

direction is measured from the rightward horizontal

Calculating Projectile Motion - Range

• Use your kinematic equations!

• Range is your horizontal displacement - ∆d

• Remember vx is a constant velocity, so we use:

total

xx t

dv

Projectile Motion – Total Time

• Once you hit the ground, your motion stops!

• Time connects your vertical motion with your horizontal motion.

To Calculate Projectile Motion…• We use the kinematic equations• Remember… Δd = vit + ½ at2

– RANGE (x-displacement) uses x-components and total time• Δdx = vixttotal + ½ axttotal

2 = vixttotal

• Remember that a = 0 in the x-direction– HEIGHT (y-displacement) uses y-components• Δdy = viyt + ½ ayt2

• Remember that a = -9.8 m/s2 in the y-direction

More on Calculations• ay = (vfy

2 – viy2) / 2Δdy

– (to find max height)

• ay = (vfy – viy) / t – (to find time to max height)

• These only apply in the y-direction (vx doesn’t change)

• ay = -9.8 m/s2 ALWAYS

• Remember, at the top of its path, a projectile’s vy = 0 (so vfy = 0)

EXAMPLES

• A cannonball is fired from a cannon at an angle of 32° and an initial velocity of 54 m/s. A. What are the components of the initial velocity?B. How long does it take the cannonball to reach

maximum height?C. What is the cannonball’s maximum height?D. What is the total time of travel?E. What is the cannonball’s range?

A cannonball is fired from a cannon at an angle of 32 ° and an initial velocity of 54 m/s.

A. What are the components of the initial velocity?

A cannonball is fired from a cannon at an angle of 32 ° and an initial velocity of 54 m/s.

B. How long to maximum height?

A cannonball is fired from a cannon at an angle of 32 ° and an initial velocity of 54 m/s.

C. What is the maximum height?

A cannonball is fired from a cannon at an angle of 32 ° and an initial velocity of 54 m/s.

D. What is the total time?

A cannonball is fired from a cannon at an angle of 32 ° and an initial velocity of 54 m/s.

E. What is the range?