Unit 3: Projectile Motion

I. Horizontal Projectiles What is a Projectile? Mythbusters (3 min) An object that has only the force of gravity acting on it All horizontally fired objects, regardless of speed, will land at the same time if fired from the same height

Objects of different masses fall at the same rate Horizontally fired objects fall at the same rate as objects dropped.

Mythbusters Video – Ball Shot from moving car B. Properties of Horizontal Projectile Motion: Horizontal projectile motion is free fall with an initial horizontal velocity Horizontal and vertical motions are completely _____________ of each other. independent Mythbusters Video – Ball Shot from moving car

Changing (9.81 m/s each sec.) B. Properties of Horizontal Projectile Motion: Horizontal Motion (X Axis) Vertical Motion (Y Axis) Forces Acceleration Velocity Gravity, acts down None g = -9.81 m/s/s acts down None Constant vix = vfx Changing (9.81 m/s each sec.)

1 s -> 2 s -> 3 s -> 4 s -> 5 s -> B. Properties of Horizontal Projectile Motion: 1 s -> If no gravity 2 s -> 3 s -> Parabolic Trajectory (actual path) 4 s -> 5 s -> Vertical Motion Only

B. Properties of Horizontal Projectile Motion:

C. Projectile Motion Problem Solving 1. Draw a diagram and identify which directions (x an y) you want to be positive (we always use ________and to the ______________ is positive) up right 2. Create a horizontal (x) and vertical (y) table and break all given values and what you want to find into these components.

C. Projectile Motion Problem Solving 3. Remember general information: A) ay = g = -9.81 m/s/s - the projectile experiences a constant downward acceleration B) ax = zero - the object does not accelerate in the horizontal component (vi, x = vf, x) C) If fired horizontally, viy = 0 m/s

D. Horizontal Projectile Examples 1. A physics student from Pittsford throws a ball thrown horizontally from the top of a building with a speed of 8.00 m/s. The student, wishing to test her knowledge, measures the height of the building to be 15.0 m. What is the ball’s acceleration in x and y components? B) What is the ball’s initial vertical (y) velocity? C) How long does it take for the ball to hit the ground? D) How far away did the ball land from the building?

2. A plane drops cargo on a friendly village 2. A plane drops cargo on a friendly village. The plane is traveling horizontally at a constant speed. The cargo takes 20 s to hit the ground and lands 4,800 m away from where it was originally dropped (Ball shot from moving cart video (15 s)) A) What was the plane’s altitude when it released the cargo? B) Determine the speed of the plane when it dropped the cargo.

E. Horizontal Projectile WB Problems 1) You are at a circus and you see a stunt man climb up 20 meters into a cannon. He gets fired horizontally out of the cannon and lands 150 m away. A) How long was stunt man in the air for? B) What was his velocity when he came out of the cannon? 2) An arrow is shot at a target 20 m away. The arrow is shot with a horizontal velocity of 60 m/s. A) How long is the arrow in the air for? B) How far does the arrow drop as it approaches the target?   3) A stone is thrown off a cliff with a horizontal speed of 15 m/s. A) If the stone’s final vertical speed is 40 m/s, how tall was the cliff? B) How long is the stone in the air for? C) How far from the base of the cliff does the stone hit the ground? D) What is the stone’s horizontal speed right before it hits the ground? 1A) 2.01 s, B) 75 m/s 2A) 0.33 s, B) 0.53 m 3A) 81.5 m, B) 4.08s, C) 61.2 m, D) 15 m/s

II. Projectiles at Angles A. Vector Components Equations (see reference tables): Vi = magnitude of initial velocity Viy = initial y comp. of velocity vix = vi cos θ θ (Ax = A cos θ) viy = vi sin θ Vix = initial x comp. of velocity (Ay = A sin θ)

A. Vector Components Examples: If the initial velocity of an object is 30 m/s at 30 degrees. What are the initial horizontal and vertical components of the velocity?

A. Vector Components Examples: 2. If the initial horizontal velocity of an object is 22 m/s and it is fired at an angle of 13 degrees, what is its initial speed?

A. Vector Components Examples: 3. If the initial vertical velocity of an object is 100 m/s and it is fired at an angle of 15 degrees, what is its initial speed?

B. Properties of Projectiles Fired at an Angle Trajectory – Parabolic, caused by the vertical force of gravity The projectile travels with constant horizontal velocity and constant vertical acceleration At the maximum height, vertical velocity equals zero

B. Properties of Projectiles Fired at an Angle Shape of that path for a projectile fired at an angle:

B. Properties of Projectiles Fired at an Angle

B. Properties of Projectiles Fired at an Angle Horizontal (x) Motion Vertical(y) Motion Horizontal Velocity: Remains constant vix = v cosθ Vertical Velocity: changes viy = v sinθ Vertical Accel.: Equals g (ay = -9.81 m/s2) Horizontal Accel.: ax = 0 m/s2 Only equation: d = vixt Maximum Height: vy = 0 m/s

C. Projectiles at Angles Example A tennis ball is lobbed at 30 to the horizontal at +10 m/s from a height of 0 m. What is the initial velocity in the horizontal and vertical directions? B) How long does the tennis ball to reach the maximum height in its path? C) What is the total time the ball was in the air for?

C. Projectiles at Angles Example A tennis ball is lobbed at 30 to the horizontal at +10 m/s from a height of 0 m. D) What distance (range) does the tennis ball travel? E) What is the ball’s maximum height? F) If the angle of the ball decreased, what will happen to 1) flight time, 2) range, 3) height?

D. Projectiles at Angles Whiteboard A 5 kg cannon ball is fired with a velocity of 75 m/s at an angle of 40o from the horizontal. A) Determine the horizontal and vertical components of the initial velocity. B) What is the “hang time” of the cannon ball? C) What is the range of the cannon ball? D) What is the cannon ball’s maximum height above the ground? E) What is the cannon ball’s speed at its maximum height? F) What are the cannon ball’s horizontal and vertical components of its velocity 3 seconds in to the flight? G) What is the acceleration 1 second into its flight? Vx = 57.5 m/s, Vy = 48.2 m/s t = 9.83 s dx = 565m dy = 118 m v = 57.5 m/s Vx = 57.5 m/s, Vy = 18.8 m/s ax = 0 m/s/s, ay = -9.81 m/s/s

Video Clip and Gizmo Simulation You are a vet trying to shoot a tranquilizer dart into a monkey hanging from a branch in a distant tree. You know that the monkey is very nervous, and will let go of the branch and start to fall as soon as your gun goes off. On the other hand, you also know that the dart will not travel in a straight line, but rather in a parabolic path like any other projectile. In order to hit the monkey with the dart, where should you point the gun before shooting? Video Clip and Gizmo Simulation 1 Above 2 Right at him 3 Below

Shooting the Monkey... monkey! At an angle, still aim at the monkey! Dart hits the monkey!

E. Symmetry in Angles for Projectiles Golf Range Gizmo Greatest range Horizontal Distance (m) Angle (degrees)

E. Symmetry in Angles for Projectiles Conclusions: Projectiles fired at 45 degrees gives maximum horizontal range. Projectiles fired at complementary angles (add up to 90 degrees) give the same range

34.3 m/s 60.1 m 52.5 m First find: Vx = 75 m/s, Vy = 130 m/s 861 m 13.3 s   F. Extra Problems 1. A boulder rolls off a cliff with a speed of 15 m/s. A. If the boulder falls for 3.5 s and hits the ground, what is the boulder’s final vertical velocity? B. What is the height of the cliff it rolled off at? C. How far away did the boulder travel horizontally during its flight?   2. A projectile is fired at an angle of 60 degrees at 150 m/s on level ground. A. What was the maximum height of the projectile? B. How long did it take to reach its maximum height? C. What was the projectiles total time in the air? D. What was the range of the projectile? 26.6 s 1,995 m