2 – Dimensional Kinematics PROJECTILE MOTION (Right-click to pause presentation at any time)

PROJECTILE MOTION The trajectory (path) of a projectile is parabolic. The trajectory (path) of a projectile is parabolic. The vertical motion determines the time of flight.

PROJECTILE MOTION V0V0 V 0Y V 0X V1V1 V 1X = V 0X V 1Y = 0 m/s V2V2 V2V2 V 2Y V 2X = V 0X The horizontal motion is uniform. (a X = 0 m/s 2 ) Vertical motion is accelerated. (a Y = g = 9.80 m/s 2 ) g

SPECIAL CASE A Projectile fired horizontally, θ = 0 o VoVo ΔdyΔdy Δd x (range)

Case A Horizontal motion (uniform) Horizontal motion (uniform) NOTE : SINCE θ = 0 o, cos θ = 1 We need to use the vertical motion to find Δt

Case A Vertical motion (to find Δt) Since θ = 0 o and v oy = v o sin θ, sin θ = 0 and v oy =0 (time of flight)

Case A Pluginto HOTLINK 1

SPECIAL CASE B Projectile fired from “ground” level, Δd y = 0 m VoVo V oy g V ox ΔdxΔdx

Case B Horizontal motion (uniform) We need to use the vertical motion to find Δt We need to use the vertical motion to find ΔtΔt

Case B Vertical motion (to find Δt) Since Δd = 0 m

Case B (time of flight)

Case B Plug into HOTHOT LINK 2LINK

Case C The most general case. ΔdyΔdy VoVo VoVo ΔdyΔdy

Case C Vertical motion ( to find the time of flight) Put into standard quadratic format And apply the quadratic formula

Case C Simplifying, we get : And since v oy = v o sinθ And since v oy = v o sinθ (This is the equation for the time of flight)

Case C Horizontal motion (to find the range) Since v ox = v o cos θ, Now plug the expression for Δt into this equation.

Case C The use of “+” or “-” depends on the path of the projectile. The use of “+” or “-” depends on the path of the projectile. 1) If the projectile passes over the apex, use the “-” sign. 2) If it does not, use the “+” sign. {Since g is negative, use of the “-” sign gives a greater time of flight.} {Since g is negative, use of the “-” sign gives a greater time of flight.}

Case C For ease of calculation, this is usually written: Use “+” sign for both (not over apex) Use “+” sign for both (not over apex) Use “-” sign for both (over the apex) Use “-” sign for both (over the apex) HOTLINK 3

Homework Problem A basketball player shoots a basket by launching the ball at an angle of 60.0 o from a location 1.0 m below the rim at 12 m/s. How far is he from the basket if he makes the shot? (hint : define up as +)

Practice the problems on the Projectile Motion Practice Sheets The following pages contain all of the practice problems.

CHAPTER 7PROJECTILE MOTION PROBLEMS 1. A ball falls from rest from a height of 490 m. a. How long does it remain in the air? b. If the ball has a horizontal velocity of 2.00 m/s when it begins its fall, what horizontal displacement will it have? 2. An archer stands 40.0 m from the target. If the arrow is shot horizontally with a velocity of 90.0 m/s, how far above the bull’s-eye must he aim to compensate for gravity pulling his arrow downward? 3. A bridge is m above a river. If a lead-weighted fishing line is thrown from the bridge with a horizontal velocity of 22.0 m/s, how far has it moved horizontally when it hits the water? 4. A beach ball, moving with a speed of m/s, rolls off a pier and hits the water 0.75 m from the end of the pier. How high above the water is the pier?

5. Carlos has a tendency to drop his bowling ball on his release. Instead of having the ball on the floor at the completion of his swing, Carlos lets go with the ball 0.35 m above the floor. If he throws it horizontally with a velocity of 6.3 m/s, what distance does it travel before you hear a “thud”? 6. A discus is released at an angle of 45° and a velocity of 24.0 m/s. (assume Δdy = 0 m) a. How long does it stay in the air? b. What horizontal distance does it travel? 7. A shot put is released with a velocity of 12 m/s and stays in the air for 2.0 s. (assume Δdy = 0 m) a. At what angle with the horizontal was it released? b. What horizontal distance did it travel? 8. A football is kicked at 45° and travels 82 m before hitting the ground. a. What was its initial velocity? b. How long was it in the air? c. How high did it go?

9. A golf ball is hit with a velocity of 24.5 m/s at 35.0° above the horizontal. Find : a. the range of the ball. b. the maximum height of the ball. 10. A car moving at 120 Km/hr on a flat horizontal road loses control and careens off a cliff 75 m high into the valley below. Neglecting wind resistance, how far from the base of this sheer cliff does the car land? 11. You are a detective investigating an accident similar to the one described in problem #1 except that the cliff is 50 m high and the car impacts with the ground 20 m from the base. The speed limit was 50 Km/hr. How fast was the car going when it left the cliff? Could excess speed have contributed to the accident? 12. A baseball is struck with a bat at a height of 1.0 m giving it a speed of 60 m/s at an angle of trajectory of 30 o. How far from home plate would the outfielder have to be in order to catch the ball at a height of 1.0 m? (Note: the distance to the outfield wall in this park is 110 m.)

13. A basketball player attempts to shoot a basket from mid-court (13 m from the basket) by making a jump shot and releasing the ball from a height of 3.0 m. If the basket is 3.0 m high and he gives the ball an angle of trajectory of 50 o, at what speed must he throw the ball in order to score? 14. A bullet having a muzzle velocity of 150 m/s is fired horizontally from a height of 1.5 m. How much time passes before it strikes the ground? What is its range? 15. A cliff diver in Acapulco launches himself at 1.5 m/s at an angle of 15 o above the horizontal from a cliff 60 m above the surf. In order to survive, he must clear the jagged cliffs with a horizontal displacement of 10 m. Will he live to dive again?

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