What is Projectile Motion?

Projectile Motion

• Two-dimensional motion of an object– Vertical– Horizontal

Projectile Motion

The ball is in free fall vertically and moves at constant speed horizontally!!!

v0

x

y

x

y

x

y

x

y

x

y

x

y

•Motion is accelerated

•Acceleration is constant, and downward

• a = g = -9.81m/s2

•The horizontal (x) component of velocity is constant

•The horizontal and vertical motions are independent of each other, but they have a common time

g = -9.81m/s2

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

Projectile Motion

Examples of Projectile Motion

• Launching a Cannon ball

Projectile Motion

x

yHorizontal (x)

x v x t

Vertical (y)

y y0 v0y t 12gt 2

v y v0y gt

Problem 1:

• Two tennis ball launchers shoot balls at the same time, angle and initial speed from different floors of a tall building.

• The balls land in the street below. Ignore air resistance.

Problem 1:

• a. Which ball will have the greater acceleration while in flight? Explain your reasoning.

• Both have the same vertical acceleration since the net force is due to gravity alone (there is no horizontal acceleration.) Even if the balls were of different mass the acceleration would be the same.

Problem 1:

• b. Which ball will hit farther from the base of the building? Explain your reasoning.

• Ball A will hit farther from the building since it has a higher starting position and will therefore be in the air longer before it hits the ground. The longer it is in the air, the more time it has to move horizontally.

Problem 1:

• c. Which ball will reach a greater maximum height? Explain your reasoning.

• Ball A will reach a greater maximum height since its launcher is above B's launcher.

Problem 1:

• d. Which ball will be going faster just before hitting the street? Explain your reasoning.

• Ball A will be going faster. Both have the same x-velocity, but A will have a greater y-velocity since it has accelerated from a higher height.

Problem 2:

1. A roadrunner runs directly off a cliff with an initial velocity of 3.5 m/s. What are the components of this velocity?

Answer: Vx = 3.5 m/s Vy = 0 m/s

2. What will be the horizontal velocity 2 seconds after the bird leaves the cliff?

Answer: 3.5 m/s – horizontal velocity is unchanging

3. If the cliff is 300 m high, at what time will the roadrunner reach the ground? (v0 and d0 are both zero)

h = dy = ½ * 10 * t2 = 300 t2 = 300 * 2 / 10 = 60t = 7.75 s

Horizontal (x)

x v x t

Vertical (y)

y y0 v0y t 12gt 2

v y v0y gt

Problem 2:

4. How far from the cliff will this bird land?

• dx = 3.5 * 7.75 = 27.125 m5. What is the final vertical velocity at which the

roadrunner is traveling? [The vertical velocity at the time when the bird reaches the ground]

• Vy = 0 + 9.8 * 7.75 = 77.5 m/s6. What is the final horizontal velocity at which the

roadrunner is traveling? [The horizontal velocity at the time when the bird reaches the ground]

• Vx = 3.5 m/sHorizontal (x)

x v x t

Vertical (y)

y y0 v0y t 12gt 2

v y v0y gt

Projectile Motion and Satellites

Satellites

• Curvature of the earth enters into our calculations

If I start 5 m above the surface, it will still be at 5 m after one second if it is moving 8000 m/sec

Satellites

Throw at 8000 m/sec

This is about 18,000 mph

Earth circumference is 25,000 miles

Takes 25000/18000 = 1.4 hours = 84 minutes

Higher altitude longer

Satellites

Force of gravity on bowling ball is at 90o to velocity, so it doesn’t change the velocity!!!

If no air resistance, gravity doesn’t change speed of satellite, only direction!!!

Communications Satellites

• Farther out you go, the bigger the circumference of the orbit

• It takes longer for the trip

• Also, gravity weakens by inverse square law the farther out you go

• Make the distance so that it takes 24 hours for the orbit

• Satellite is stationary in the sky!!!

The Moon

• Distance is about 240,000 miles

• Takes 27.3 days to make an orbit

Elliptical Orbits

• Give the object a speed a bit greater than 8 km/sec and the orbit will be elliptical

Elliptical Orbits

Elliptical Orbits

Sum of distances from foci to point on the ellipse is a constant!!!