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10. Free fall Free fall acceleration: g=9.8m/s 2 Using general equations: Substitute: To derive the following equations: 1

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Example: A rocket is fired at a speed of 100 m/s straight up. (Neglect air resistance.) a)How long does it take to go up to the highest point? b)What is the maximum height? c)How long does it take to return back (go up and fall back down)? d)What is the speed of the rocket just before it falls down? Solution: a) At the highest point c) When it comes back Given: b) The maximum height is d) The speed it falls down with is 2

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The trajectory of an object projected with an initial velocity at the angle above the horizontal with negligible air resistance. 11. Projectile Motion 3

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Shoot the monkey (tranquilizer gun) A zookeeper shoots a tranquilizer dart to a monkey that hangs from a tree. He aims at the monkey and shoots a dart with an initial speed v 0. The monkey, startled by the gun, lets go immediately. Will the dart hit the monkey? A.Only if v 0 is large enough. B.Yes, regardless of the magnitude of v o. C.No, it misses the monkey. If there is no gravity, the dart hits the monkey… If there is gravity, the dart also hits the monkey! 4

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If there is no gravity, the dart hits the monkey… Continued 5

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If there is gravity, the dart also hits the monkey! Note, that it takes the same amount of time to hit the monkey as in the no gravity case! Continued 6

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This might be easier to think about… For the bullet: For the monkey: Continued 7

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Example: A rocket is fired at a speed of 100 m/s at an angle 30 ° above the horizontal. (Neglect air resistance.) a)What are the initial values of the x and y components of the speed? b)How long is the rocket be in the air? c)What is the distance between the lunch and landing points (assuming that these points have the same altitude)? Solution: Given: a) b) From the example on slide 2, question (c) follows that c) The horizontal motion is uniform, and the distance is 8

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12. Circular motion Definitions: For any circular motion there is radial (centripetal) acceleration: It is directed along radius and to the center. For uniform circular motion ( v=const ): T – period (time it takes to return to the same point) 9

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Optional information (if you are curious): If circular motion is not uniform ( v ≠ const ) then there is tangential acceleration : The tangential acceleration is perpendicular to the radial acceleration, and the total acceleration is equal to: 12a. Circular motion (continues) 10

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Example: Period of a satellite motion g R 11

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Example: Two balls attached to a string as shown at 0.20 m and 0.40 m from the center move in circles at a uniform frequency of 20 rpm. What are their periods, linear speeds, and radial accelerations? 12

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Chapter 4 MOTION IN TWO DIMENSIONS. Two dimensions One dimension Position O x M x x y M Path of particle O x y.

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