QOD

• The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a rock horizontally off the bridge. The rock hits the water such that the magnitude of its horizontal displacement is 45.0 m. Find the speed at which the rock was kicked.

What is Projectile Motion?

Example

• Every March, the swallows return to San Juan Capistrano, California after their winter in the south. If the swallows fly due north and cover 200 km on the first day, 300 km on the second day, and 250 km on the third day, draw a vector diagram of their of their trip and find their total displacement for the three-day journey.

Example

• Suppose a car pulling forward with a force of 20,000 N was pulled back by a rope that Joe held in his teeth. Joe pulled the car with a force of 25,000 N. Draw a vector diagram of the situation and find the resultant force.

Projectiles

• Any object that moves through the air or space, acted on only by gravity– Near Earth, follow a curved path– Can be broken down into two components.

• Vertical• Horizontal

Horizontal Component

• Like horizontal motion of a ball rolling freely on a level surface

• Friction is negligible, rolling ball moves at a constant velocity

• Ball covers = distance in = time intervals• If no horizontal force acts on the ball, there

is not horizontal acceleration• Same is true for projectiles.

“When no horizontal force acts on a projectile, the horizontal velocity remains

constant.”

Example

• If New York Yankee, Mark Texeria hits a baseball due west with a speed of 50.0 m/s, and the ball encountered a wind that blew it north at 5.00 m/s, what was the resultant velocity of the baseball?

Example

• The Maton family begins a vacation trip by driving 700 km west. Then the family drives 600 km south, 300 km east, and 400 km north. Where will the Matons end up in the relation to their starting point. Solve graphically.

Example

• Ralph is mowing the backyard with a push mower that he pushes downward with a force of 20.0 N at an angle of 30.0o to the horizontal. What are the horizontal and vertical components of the force exerted by Ralph

Homework

1. Some Antarctic explorers heading due south toward the pole travel 50. km during the first day. A sudden snow storm slows their progress and they move only 30. km on the second day. With plenty of rest they travel the final 65 km the last day and reach the pole. What was the explorers’ displacement?

2. Erica and Tony are out fishing on the lake on a hot summer day when they both decide to go for a swim. Erica dives off the front of the boat with a force of 45 N, while Tony dives off the back with a force of 60. N. a) Draw a vector diagram of the situation. b) Find the resultant force on the boat.

3. Derek pulls a horses’ bridle from the front with a force of 200.N and Dan pushes the horse from behind with a force of 300. N. a) Draw a vector diagram of the situation. b) What is the resultant force on the horse?

4. Shareen finds that when she drives her motorboat upstream she can travel with a speed of only 8 m/s, while she moves with a speed of 12 m/s when she heads downstream. What is the current of the river?

Homework (con’t)

5. Amit flies due east from San Francisco to Washington, D.C., a displacement of 5600. km. He then flies from Washington to Boston, a displacement of 900. km at an angle of 55.0o east of north. What is Amit’s total displacement?

6. Marcie shovels snow after a storm by exerting a force of 30.0 N on her shovel at an angle of 60.0o to the vertical. What are the horizontal and vertical components of the force exerted by Marcie?

Vertical Component

• Resembles that of free falling objects.

• Force due to gravity in vertical direction

• Projectile accelerated downwards

• Velocity changes with time v = at

• Increased speed in vertical direction causes greater distance to be covered with each successive time interval

• Horizontal Component TOTALLY independent from vertical component.

• Combined effects generate the curved paths projectiles follow.

Horizontal + Vertical = Curved Path

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

Simultaneous Balls Dropped

• One ball launched horizontally

• One ball simply dropped

• Analyze the path by considering horizontal and vertical components separately

Horizontal

• Ball moves same horizontal distance in = time intervals

• Happens because no horizontal component of force acts on it.

• The balls horizontal component of motion remains the same

Vertical

• Ball’s vertical component of motion remains the same

• Both balls fall the same vertical distance in the same timeVertical distance fallen

has nothing to do with horizontal component of acceleration

Downward motion of horizontally projected balls is the same as free fall

Examples of Projectile Motion

• Launching a Cannon ball

QOD

• A hiker walks 25.5 km from her base camp at 35o south of east. On the second day, she walks 41.0 km in a direction 65o north of east, at which point she discovers a forest ranger’s tower. Determine the magnitude and direction of her resultant displacement between the base camp and the ranger’s tower.

Example

• In her physics lab, Melanie rolls a 10 g. marble down a ramp and off the table with a horizontal velocity of 1.2 m/s. The marble falls in a cup placed 0.51m from the table’s edge. How high is the table?

Example

• Bert is standing on a ladder picking apples in his grandfather’s orchard. As he pulls each apple off the tree, he tosses it into a basket that sits on the ground 3.0 m below at a horizontal distance of 2.0 m from Bert. How fast must Bert throw the apples (horizontal) in order for them to land in the basket?

• Path traced by a projectile accelerating only in the vertical direction while moving at constant horizontal velocity is called a parabola.

Upwardly Launched Projectiles

• Cannonball follows curved path and hits the ground because of gravity.

• If no gravity, cannonball follows a straight line path shown by dotted line.

Because of Gravity…

• Cannonball continually falls beneath imaginary line until it hits ground

• Vertical distance it falls beneath any point on dashed line is same vertical distance it would fall if it dropped from rest and had fallen for the same amount of time.

• Distance can be measured by d = ½ at2

• Since there is no horizontal acceleration, the cannonball moves = horizontal distances per time interval.

Example

• A famous human cannonball of the Ringling Bros. and Barnum & Bailey Circus, was fired out of a cannon with a speed of 24.0 m/s at an angle of 40.0o to the horizontal. If he landed in a net 56.5 m away at the same height from which he was fired, how long was he in the air?

Example

• On May 20, 1999, Robbie Knievel successfully jumped 69.5 m over a Grand Canyon gorge. Assuming that he started and landed at the same level and was airborne for 3.66 s, what height from his starting point did he achieve?

QOD

• Abby Brown hates baseball, so she kicks a soccer ball at an angle of 25.0o relative to the ground at a speed of 23.0 m/s. If the ball was stopped by the goalie 42.0 m away, how long was it in the air? How high was the tallest spot in the ball’s path?

Vectors

• Vectors represent both vertical and horizontal components for a projectile on a parabolic path.

• Horizontal component always the same

same magnitude/value

• Resultant velocity vector represented by rectangle formed by component parts.

• Velocity at top of path = horizontal component because vertical velocity is zero

Factors Affecting Projectile Motion

• What two factors would affect projectile motion?– Angle– Initial velocity

Initial Velocity

Angle

Projectiles Shot at Steeper Angles

• Initial velocity has greater vertical component than when projectiles angle is less

• Greater vertical component results in higher path

• Horizontal component is less, therefore the range is less

Things to Note

• Same range obtained for two different projectile angles – angles that add up to 90 degrees.

• Thrown at 60 degrees, will land in the same place as object thrown at a 30 degree angle with same speed.

• Smaller angle remains in air for shorter amount of time.

Air Resistance

• If air resistance is not negligible, range of projectile is smaller and path is not a true parabola.

• If there is no air resistance…

1)Projectile rises to maximum height in some time in same time it takes to fall from height to ground.

2)Deceleration due to gravity going up = acceleration due to gravity coming down

3)Object hits ground at the same speed it had originally left ground

Fast Moving Objects

• If an object were thrown so fast that is clears the horizon, it would then orbit the Earth and become an Earth Satellite.

• Earth Satellite – projectile moving fast enough to fall around the earth rather than into it.

Class Exercise

An object is fired from the ground at 100. meters per second at an angle of 30.0o degrees with the horizontalCalculate the horizontal and vertical components of

the initial velocityWhat is the maximum height achieved?What is the maximum distance achieved?

After 2.00 seconds, how far has the object traveled in the horizontal direction?

How high is the object at this point?

Solution

• Part a

• Part b

• Part c

mssmtvx

t

xv

x

ix

1740.287

s

ms

mvv

sm

smvv

iiy

iix

5030sin100sin

8730cos100cos

0

0

222 0.28.9

2

10.250

2

1s

smss

mtgtvy iy

Applications

Any Ideas?

LAB TIME!!!