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acceleration Simplest case a=constant. Equations hold even if Δt large. Δv =vf -vi ti= 0

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**Example : If a car traveling at 28 m/s is brought to a full stop 4**

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: vi = +28 m/s, vf = 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 Vav gives same area Hence same distance

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**Constant acceleration**

Δx= vavt Δx= = vit+1/2 at2 ti = 0

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Fig

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**viΔt + ½ aΔt2 Δx=vxΔt=Area = viΔt (blue area)+ ½ Δt (aΔt) (gold area)**

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**a=constant (vf2-vi2) =2aΔx Δx=vavΔt = 1/2(vi+vf)Δt**

but vf = vi + aΔt so Δt = (vf-vi)/a Δx = 1/2(vi+vf)Δt = 1/2 (vi+vf) (vf-vi)/a = 1/(2a) (vf2-vi2) = Δx (vf2-vi2) =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/s2. 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. vf = vi + aΔt (vf2-vi2) =2aΔx

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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/s2. 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/s2 The mass of the object doesn’t matter! Heavy and light objects all fall with the same g It doesn’t 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 at2 Substitute the numbers a=9.8 t=2 Δx=19.6 How fast is it going when it hits?

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Example Problem 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? Slide 2-34

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**. v(4) =0. It reaches its highest point when it stops**

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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**. v(4) =0. It reaches its highest point when it stops**

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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**Checking Understanding**

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 arrow’s acceleration at the different points? A E B D; C 0 E D C B A A B C D E A B D E; C 0 Answer: C Slide 2-37

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Answer 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 arrow’s acceleration at the different points? A E B D; C 0 E D C B A A B C D E A B D E; C 0 Answer: C Slide 2-38

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**Checking Understanding**

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 2-39

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Answer 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 D. Slide 2-40

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**Given: viy = +15.0 m/s; ay = 9.8 m/s2**

Example: You throw a ball into the air with speed 15.0 m/s; how high does the ball rise? Given: viy = m/s; ay = 9.8 m/s2 x y viy ay

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**(vf2-vi2) =2ad Also an how far question**

What is vf? vi? What is a? (be careful of signs!)

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**You toss a ball straight up with an initial vi=25m/s**

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

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**Given: viy = 0 m/s, ay = 9.8 m/s2, y = 369 m**

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? x y Given: viy = 0 m/s, ay = 9.8 m/s2, y = 369 m 369 m ay Unknown: vyf

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Unit 3 Kinematics Equations

Unit 3 Kinematics Equations

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