Projectile Motion

What is a Projectile?

• Any object that continues in motion by its own inertia and is influenced only by the downward force of gravity (and air resistance – which we will ignore)
• an object dropped from rest
• an object launched straight upward
• an object launched horizontally
• an object launched at an angle

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What is a Projectile?

• Since a projectile moves in 2-dimensions it has 2 components just like a resultant vector
• Horizontal and Vertical

Horizontal Component

• NEVER changes, covers equal displacements in equal time periods
• This means the initial horizontal velocity equals the final horizontal velocity

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In other words, the horizontal velocity is CONSTANT. BUT WHY?

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Gravity DOES NOT work horizontally to increase or decrease velocity

Vertical Component

• Changes (due to gravity), does NOT cover equal displacements in equal time periods

• Both the MAGNITUDE and DIRECTION change
• As the projectile moves up the MAGNITUDE DECREASES and its direction is UPWARD
• As it moves down the MAGNITUDE INCREASES and the direction is DOWNWARD

Combining the Components

• Together, these components produce what is called a trajectory or path
• This path is parabolic in nature

Particle Models

Horizontally Launched Projectiles

• Projectiles which have NO upward trajectory and NO initial VERTICAL velocity

v_iy = 0 m/s

v_ix = v_x = constant = 20 m/s

Horizontally Launched Projectiles

• Which equations do we use?
• Any kinematics equation can be used BUT a separate equation must be used for each component
• The velocity is CONSTANT horizontally, so that means the acceleration is ZERO
• Since the projectile is launched horizontally, the INITIAL VERTICAL VELOCITY is equal to ZERO

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∆x = v_it + ½ at²

v_f = v_i + at

v_f² = v_i² + 2a∆x

Horizontally Launched – Example 1

You throw a baseball from the roof of a house to a friend on the ground. The ball has an initial velocity of 12 m/s in the horizontal direction. After 1 second, how fast is the ball moving in

(a) the horizontal direction

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(b) the vertical direction

v_fx = v_ix = 12 m/s

v_fy = v_iy + at

v_fy = 0 + (-9.8)(1)

v_fy = -9.8 m/s

v_x = 12 m/s

t = 1 s

v_iy = 0 m/s

Horizontally Launched – Example 1

(c) What is the speed of the ball after 1 second?

Resultant² = v_x² + v_y²

c² = a² + b²

c² = 12² + 9.8²

c = 15.5 m/s

12 m/s

-9.8 m/s

Vertically Launched Projectiles

Horizontal Velocity is constant

Vertical Velocity decreases on the way upward

Vertical Velocity increases on the way down

NO Vertical Velocity at the top of the trajectory

Vertically Launched Projectiles

Since the projectile was launched at an angle, the velocity MUST be broken into components

v_i

v_ix

v_iy

θ

Vertically Launched Projectiles

• Notes:
• If it begins and ends at ground level, the “y” displacement ∆x_y is ZERO
• At the top of the trajectory, v_y = 0, but v_x does not

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Vertically Launched – Example 2

A place kicker kicks a football with an initial vertical velocity of 49 m/s and an initial horizontal velocity of 3 m/s.

(a) What is its horizontal velocity after 5 seconds?

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(b) What is its vertical velocity after 5 seconds?

3 m/s

0 m/s

v_iy = 49 m/s

v_x = 3 m/s

a = -9.8 m/s²

v_fy = v_iy + at

v_fy = 49 + (-9.8)(5)

v_fy = 0 m/s