Motion in Two Dimensions

Projectile Motion: the motion of a particle that is projected or launched and only accelerated

by gravity.

cp: 5

Parabolic Motion of Projectiles: the horizontal motion and the vertical motion are independent of each other.

sin

cos

ooy

oox

vv

vv

Projectile Motion

• Horizontal and Vertical Motions are “separate”

• Horizontal : constant velocity (ignoring air resistance) x=vot

• Vertical: constant acceleration 4 Motion equations for constant acc.

See M&M movies

The Plane and The Package

The Truck and The Ball

Horizontal Range

tvR

xxRRange

o

i

)cos(

)(

Note: this does not give the horizontal distance traveled by a projectile when the final height is not the launch height!

(see example on pg. 159)

The Monkey and Zookeeper

There is an interesting monkey down at the Glenview Zoo. The monkey spends most of its day hanging from a limb of a tree. The zookeeper feeds the monkey by shooting bananas from a banana cannon to the monkey in the tree. This particular monkey has a habit of dropping from the tree the moment that the banana leaves the muzzle of the cannon. The zookeeper is faced with the dilemma of where to aim the banana cannon in order to hit the monkey. If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon?

The Monkey and Zookeeper: Earth gravity(aim above monkey)

The Monkey and Zookeeper: Earth gravity(aim at the monkey and fire banana fast)

The Monkey and Zookeeper: Earth gravity(aim at the monkey and fire banana slow)

Practice problems1) A stone is thrown horizontally at 15 m/s from the top of a cliff 44 m high.A) How far from the base of the cliff does the stone hit the ground?B)How fast is it moving the instant before it hits the ground?

Practice problems

A steel ball rolls with constant velocity across a tabletop 0.950 m high. It rolls off and hits the

ground 0.352 m from the edge of the table. How fast was the ball moving?

Warm up

• How much is the acceleration of a projectile?• How are the vertical and horizontal motions related?• What quantity is the same for both the horizontal(x) and

vertical (y) motion?• By how much does the speed of an object change for

each second that it is in free fall.

• A pool ball leaves a 0.60-meter high table with an initial horizontal velocity of 2.4 m/s. Predict the time required for the pool ball to fall to the ground and the horizontal distance between the table's edge and the ball's landing location.

• B-17 is flying at 375 km/h. The bombs travel a horizontal distance of 5250 m. What was the altitude of the plane at the time they had the old "bombs away"?

• A soccer ball is kicked horizontally off a 22.0-meter high hill and lands a distance of 35.0 meters from the edge of the hill. Determine the initial horizontal velocity and vertical velocity of the soccer ball just before it hits the ground.

Warm up

• A stone is thrown horizontally at a speed of 5.0 m/s from the top of a cliff 78.4 m high.

• A) How long does it take the stone to reach the bottom of the cliff?

• B) How far from the base of the cliff does the stone hit the ground?

• C) What are the horizontal and vertical components of the stone’s velocity just before it hits the ground?

Pirate monkeys on a boat are positioned at rest 100.0 m from the base of a 40.0 m high cliff. A cannon sits on the cliff and fires horizontally at the boat. a. What must the horizontal velocity of the cannonball be to hit the boat? b. What is the velocity of the cannonball at the instant before it hits the boat?

Projectiles

• Write a definition for the following symbols:

• Vyo

• Vy

• y• a• t

• Vxo

Vertically launched projectiles

Problem solving strategies

• Launch height will always be equal to final height• Time is the same in the x and y direction• Sketch situation• Identify given/unknown

• Solve for Vyo and Vxo

• List known and unknown quantities. Remember that vx never changes, and that vy = 0 at the highest point.

• Plan how you will proceed. Use the appropriate equations; you may have to combine some of them.

Problem solving strategies

• For vertically launched projectiles use the first motion equation to find the time in the air

• Vy = Vyo + at

• At the top Vy = 0

Practice

• A ball is given an initial velocity of 22. 7 m/s at an angle of 66.0 to the horizontal.

a) How much time will it take the ball to reach maximum height?

b) How high will the ball will go?

c) What is the range of the ball?

• When chasing a hare along a flat stretch of ground, a greyhound leaps into the air at a speed of 12.5 m/s, at an angle of 25˚ above the horizontal.

• A) For how much time is he in the air?• B) What is the range of his leap?• C) How high is the dog after 1 second?

More practice

• A projectile is fired from ground level with an initial speed of 55.6 m/s at an angle of 41.2° above the horizontal.

(a) Determine the time necessary for the projectile to reach its maximum height.

(b) Determine the maximum height reached by the projectile.

(c) Determine the horizontal and vertical components of the velocity vector at the maximum height.

(d) Determine the horizontal and vertical components of the acceleration vector at the maximum height.

Horizontally launched projectile

• A penny is launched horizontally from the top of a tall building. The building is 100. m tall and the penny lands 23.6 m from the side of the building.

• a) Find the time in the air• b) Find the initial horizontal velocity.

Warm up

• An arrow is shot at 30.0 ˚ above the horizontal. Its velocity is 49 m/s and it hits the target, which is at the same height as the arrow was initially launched at.

• a) What is the maximum height the arrow will attain?• b) How far away is the target?

Horizontally launched projectile

• A football is thrown horizontally at a speed of 15.12 m/s from the top of a cliff 68.3 m high.

• A) How long does it take the football to reach the bottom of the cliff?

• B) How far from the base of the cliff does the football hit the ground?

• A naval gun fires a projectile. The gun’s muzzle velocity (so speed of the bullet) is 345 m/s at an elevation of 32.0 . What is the range of the shot?

• A punter punts a ball with an initial velocity of 196 m/s at an angle of 79.0 to the horizontal. Find how high the ball will go.