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acceleration Simplest case a=constant. Equations hold even if Δt large. Δv =v f -v i t i = 0

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Example : If a car traveling at 28 m/s is brought to a full stop 4.0 s after the brakes are applied, find the average acceleration during braking. Given: v i = +28 m/s, v f = 0 m/s, and t = 4.0 s.

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If a = const. Not true in general If a = const. one dimensional motion

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Fig V av gives same area Hence same distance

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Constant acceleration Δ x = v av t Δx= = v i t+1/2 at 2 t i = 0

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Fig

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Δx=v x Δt=Area = v i Δt (blue area)+ ½ Δt (aΔt) (gold area) = viΔt + ½ aΔt 2

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a=constant Δx=v av Δt = 1/2 (v i +v f )Δt but v f = v i + aΔt so Δt = (v f -v i )/a Δx = 1/2 (v i +v f )Δt = 1/2 (v i +v f ) (v f -v i )/a = 1/(2a) (v f 2 -v i 2 ) = Δx (v f 2 -v i 2 ) =2aΔx

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Plotted vs time

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Minimum length of runway A fully loaded 747 with all engines at full throttle accelerates at 2.6 m/s 2. Its minimum takeoff speed is 70 m/s. How long will it require to reach take off speed? What is the minimum length of a runway for a 747. (v f 2 -v i 2 ) =2aΔx v f = v i + aΔt

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Problem A car traveling at a speed of 30 m/s. A deer runs across the road and the driver slams on the brakes. It takes.75 s to begin applying the brakes. With the brakes on the car decelerates at 6 m/s 2. How far does the car travel from the instant the driver sees the deer until he stops.

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Free Fall All objects, under the influence of only gravity fall. (We are neglecting air resistance) They all fall with a constant acceleration (down) of g = 9.8 m/s 2 The mass of the object doesnt matter! Heavy and light objects all fall with the same g It doesnt matter in which direction it is moving it has an acceleration of g Since we normally take y + up free fall is -g

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Free Fall Slide 2-36

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You drop a stone off a cliff and hear it hit the ground after two seconds. How high is the cliff? This is an how far question Δx= 1/2 at 2 Substitute the numbers a=9.8 t=2 Δx=19.6 How fast is it going when it hits?

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Tennis balls are tested by measuring their bounce when dropped from a height of approximately 2.5 m. What is the final speed of a ball dropped from this height? Example Problem Slide 2-34

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Throwing stones (up) What happens if you toss a stone straight up? v(3) v(1) v(0). v(4) =0. It reaches its highest point when it stops going up, i.e. when v = 0 a is always downward it is g

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Throwing stones (up) What happens when it starts coming down again? v(3) v(1) v(0). v(4) =0. It reaches its highest point when it stops going up and then begins to fall with a=-g. It re-traces its path and velocity but down a is always downward it is g

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An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which choice below best represents the arrows acceleration at the different points? Checking Understanding A.A E B D; C 0 B.E D C B A C.A B C D E D.A B D E; C 0 Slide 2-37

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An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which choice below best represents the arrows acceleration at the different points? Answer A.A E B D; C 0 B.E D C B A C.A B C D E D.A B D E; C 0 Slide 2-38

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An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity. Checking Understanding Slide 2-39

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An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity. Answer D. Slide 2-40

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Example: You throw a ball into the air with speed 15.0 m/s; how high does the ball rise? Given: v iy = m/s; a y = 9.8 m/s 2 x y v iy ayay

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Also an how far question (v f 2 -v i 2 ) =2ad What is v f ? v i ? What is a? (be careful of signs!)

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You toss a ball straight up with an initial v i =25m/s. You then become distracted. How long until the ball clunks you on your head?

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Example : A penny is dropped from the observation deck of the Empire State Building 369 m above the ground. With what velocity does it strike the ground? Ignore air resistance. How long will it take to hit? 369 m x y Given: v iy = 0 m/s, a y = 9.8 m/s 2, y = 369 m Unknown: v yf ayay

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