Projectile Motion. Remember that the projectile travels vertically (up and down y) in the same time that it is traveling above the horizontal (x)

Transcription

1 Projectile Motion Consider motion in and y separately Ignore air resistance elocity in -direction is constant Write down positions in and y as a function of time Remember that the projectile traels ertically (up and down y) in the same time that it is traeling aboe the horizontal () The only acceleration is that due to graity, acting downward (a rocket or an object which is self propelled is not considered a projectile and does not undergo projectile motion, because it can be accelerated arbitrarily in any direction.) September 9 1

2 ŷ g is constant ˆ a a y g yˆ m g 9. 8 yˆ s In the absence of air resistance: no forces act in -direction, so, the speed in -direction is constant throughout the path. Speed changes in y-direction because of graity. September 9

3 September 9 3 t t y y Projectile motion therefore follows that of a parabola: 1 a t 1 a y t 1 gt t t t y y 1 gt 1 g y y Equation of an upside down parabola in and y

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5 Check your understanding 3.3 A projectile has elocity and acceleration. Which one or more of the figures below could not represent the directions for and at any point on the trajectory? a a A) Pictures (a) and (b) B) Pictures (c) and (d) C) Pictures (d) alone D) Pictures (a) and (c) E) Pictures (a), (c), and (d) September 9 5

6 Problem 3.15: A rock climber throws a small first aid kit to a another climber who is higher up the mountain. The initial elocity of the kit is 11 m/s at an angle of 65 degrees aboe the horizontal. At the instant when the kit is caught, it is traeling horizontally, so its ertical speed is zero. What is the ertical distance between the climbers? y 65 September 9 6

7 Problem 3.: A car dries straight off the edge of a cliff that is 54 meters high. The police at the scene of the accident obsere that the point of impact of the car is 13 meters from the base of the cliff. How fast was the car traeling when it went oer the cliff? September 9 7

8 Stones 1 and are thrown with the same speed,, but at an identical angle θ aboe and below the horizontal. Which stone hits the water with the greater speed? A) stone has the larger speed B) stone 1 has the larger speed C) not enough information D) they hae the same speed September 9 8

9 Symmetries: 1) The time required for an object to reach its maimum height, when thrown, up is equal to the time it takes for the object to return to it s starting point. ) At any displacement y aboe the point of release, the coin s upward speed equals its downward speed. September 9 9

10 Problem 3.37: An airplane is flying with a elocity of 4 m/s at and angle of 3 degrees with the horizontal, as the drawing shows. When the altitude of the plane is.4 km, a flare is released from the plane. What is the angle θ? September 9 1

11 What is the maimum height of the ball? At maimum height: y = H ; y = Find the time to reach maimum height, t ma : y = y gt y tma = y gt ma g Find H: 1 y y t gt y 1 y ma ma H t gt y 1 y g g g y sin H g g

12 What is the range of the ball? At range (maimum distance): = R; y = Find the time to reach = R : Find R: y t gt 1 y t 1 y t R t cos gt R sin g t R t (start) y Trig identity: sin cos sin g (range) y For a kickoff with = m/s and = 4, t R =.9 s R = 49 m H = 1 m sin R g g Ma R for = 45

13 Conceptual Question 3.5: A tennis ball is hit upward into the air and moes along an arc. Neglecting air resistance, where along the arc is the speed of the ball a minimum or a maimum? A) Minimum at the beginning maimum at the end B) Maimum both at the beginning and end minimum in the middle C) Maimum at the beginning minimum at the end D) Minimum both at the beginning and the end maimum in the middle E) The ball has the same speed eerywhere on its path September 9 13

14 Question: Suppose you are driing a conertible with the top down. The car is moing to the right at constant elocity. You point a rifle straight up into the air and fire it. In the absence of air resistance, where would the bullet land? (a) behind you (b) ahead of you (c) in the barrel of the rifle

15 Eample tennis players (problem 3.4): The player on the left lobs the ball with an initial speed of 15. m/s, at an angle of 5. aboe the horizontal. At this instant, his opponent is near the net, 1. m away from the ball. He begins moing away from the net.3 s later, hoping to reach the ball and hit it back at the moment that it is.1 m aboe its launch point. With what minimum aerage speed must he moe?

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17 Problem 3.46: A rifle is aimed at a small can. At the instant the rifle is fired, the can is released. Show that the bullet will always hit the can, regardless of the initial speed of the bullet. y H H D y September 9 17

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