Introduction to Mechanics Projectiles Time of Flight

Transcription

1 Introduction to Mechanics Projectiles Time of Flight Lana Sheridan De Anza College Oct 24, 2017

2 Last time height of a projectile

3 Warm Up Question # 57, page 107 Child 1 throws a snowball horizontally from the top of a roof; child 2 throws a snowball straight down. Once in flight, is the acceleration of snowball 2 (A) greater than, (B) equal to, or (C) less than the acceleration of snowball 1?

4 Overview more about the height of of projectile time of flight of a projectile

5 Height of a Projectile tely om- How can we find the maximum height that a projectile reaches above its launch point? Find the height when v y = 0. - he re c tile t i 5 oriolf O y S vi u i h R v y 0 Figure 4.9 A projectile launched over a flat surface from the origin x v 2 fy = v 2 0y 2g y 0 = v 2 0y 2gh h = v 2 0y 2g In the diagram, v 0y = v 0 sin θ. h = v 2 0 sin2 θ 2g

6 uiz Effect 4.3 Rank of changing the launch launch angles for angle the five paths in Figure 4.10 w t to time of flight from the shortest time of flight to the longest. y (m) v i 50 m/s Complementary values of the initial angle u i result in the same value of R x (m) 1 Figure from Serway & Jewett, 9th ed.

7 O FIGURE 4 12 Conceptual Exercise 5 Height and initial speed conceptual question # 6 6. Three projectiles (A, B, and C) are launched with different initial Three projectiles speeds so (A, that they B, and reach C) the are same launched maximum height, with different as shown initial speeds soin that Figure they reach List the the projectiles same maximum in order of increasing height, (a) as initial speed and (b) time of flight. Indicate a tie with an equal sign. shown. List the projectiles in order of increasing initial speed. y level from Figure 4-14 speed of the other two p (A) A, B, C (B) C, B, A (C) B, C, A (D) all the same 1 Walker, Physics, page 100. A B C O FIGURE 4 13 Conceptual Exercise 6 x 11. Child 1 thro child 2 thro acceleration less than th

8 Using the Max Height Equation 3.00 m-high tree that is 14.0 m away. The ball lands at the same level nce of 17.8 m on the green, of course. (a) If the ball left the club 54.0 what was its initial speed? (b) How high was the ball when it passed In Example 4-5 in the textbook, page 93, a golfer hits a ball over a tree onto the green. igin, r the tory n- n all Height, y (m) O 54.0 v x = d t = 17.8 m 2.24 s Distance, x (m) = 7.95 m/s The example asks, How high was the ball when it passed over the tree? If you are given the initial speed and launch angle, can you use the equation to answer the question? (A) Yes (B) No h = v 2 0 sin2 θ 2g

9 leapers (such as deer and dancers) look particularly graceful, can also make life more dangerous for to spawn. t Not Using the Max Height Equation How high was the ball when it passed over the tree? Suppose v 0 = 13.5 m/s, θ = 54.0 and tree is 14.0 m from golfer. How can we find the answer? er sends the ball over a 3.00 m-high tree that is 14.0 m away. The ball lands at the same level veling a horizontal distance of 17.8 m on the green, of course. (a) If the ball left the club 54.0 on the green 2.24 s later, what was its initial speed? (b) How high was the ball when it passed ing flight from the origin, of 54.0, and arcing over the ng the parabolic trajectory ls. h constant speed in the x velocity is simply horizonnowing v x and u, we can find the time when the ball this time into e height. Height, y (m) O Distance, x (m), d, by the time of v x = d t = 17.8 m 2.24 s = 7.95 m/s

10 leapers (such as deer and dancers) look particularly graceful, can also make life more dangerous for to spawn. t Not Using the Max Height Equation How high was the ball when it passed over the tree? Suppose v 0 = 13.5 m/s, θ = 54.0 and tree is 14.0 m from golfer. How can we find the answer? er sends the ball over a 3.00 m-high tree that is 14.0 m away. The ball lands at the same level veling a horizontal distance of 17.8 m on the green, of course. (a) If the ball left the club 54.0 on the green 2.24 s later, what was its initial speed? (b) How high was the ball when it passed ing flight from the origin, of 54.0, and arcing over the ng the parabolic trajectory ls. h constant speed in the x velocity is simply horizonnowing v x and u, we can find the time when the ball this time into e height., d, by the time of v x = d t = 17.8 m = 7.95 m/s 2.24 s Height, y (m) O Distance, x (m) Go back to the kinematics expressions! y = v 0y t 1 2 gt2 We can find the height if we know the time the ball was over the tree.

11 h Shot a golfer Not sends Using the ball over a 3.00 the m-high Max tree that is Height 14.0 m away. The Equation ball lands the same level ter traveling a horizontal distance of 17.8 m on the green, of course. (a) If the ball left the club 54.0 nded on the green 2.24 s later, what was its initial speed? (b) How high was the ball when it passed ll taking flight from the origin, angle of 54.0, and arcing over the ts along the parabolic trajectory tervals. es with constant speed in the x ent of velocity is simply horizonime. Knowing v x and u, we can u. 2t to find the time when the ball ting this time into ves the height. Height, y (m) O Distance, x (m) stance, d, by the time of v x = d t = 17.8 m = 7.95 m/s 2.24 s

12 Previous Example What was the strategy we used in the previous example?

13 Previous Example What was the strategy we used in the previous example? We found the time when the projectile was in the place we wanted to know about (above the tree). We used the time to find the other position coordinate.

14 Previous Example What was the strategy we used in the previous example? We found the time when the projectile was in the place we wanted to know about (above the tree). We used the time to find the other position coordinate. How would we solve this homework problem? #33. In a game of basketball, a forward makes a bounce pass to the center. The ball is thrown with an initial speed of 4.3 m/s at an angle of 15 below the horizontal. It is released 0.80 m above the floor. What horizontal distance does the ball cover before bouncing?

15 Time of Flight of a Projectile ion are time completely of flight time The t as time the com- from launch to when projectile hits the ground. How can we find the time of flight of a projectile? arabolic path y ty and accelera- ) nowhere (b) the v y 0 at what point are S to each other? v tile l case of projectile the origin at t i 5 O i u i rns to the same horifootballs, Assuming and golf that it is over launched a flat surface from from the the ground origin and lands on Figure 4.9 A projectile launched. the ground at theat same t i 5 0 height. with an initial velocity S h R x

16 Time of Flight of a Projectile Notice that just when striking the ground, y = 0. y = v 0y t a y t 2 0 = v 0 sin θt 1 2 gt2 Now cancel a factor of t. Warning! This will remove one solution to this equation in t. What is it?

17 Time of Flight of a Projectile Notice that just when striking the ground, y = 0. y = v 0y t a y t 2 0 = v 0 sin θt 1 2 gt2 Now cancel a factor of t. Warning! This will remove one solution to this equation in t. What is it? 1 2 gt = v 0 sin θ t flight = 2v 0 sin θ g

18 Time of Flight Example, #32 A soccer ball is kicked with a speed of 9.50 m/s at an angle of 25.0 above the horizontal. If the ball lands at the same level from which it was kicked, how long was it in the air? 1 Walker, Physics, page 106.

19 Time of Flight Example, #32 A soccer ball is kicked with a speed of 9.50 m/s at an angle of 25.0 above the horizontal. If the ball lands at the same level from which it was kicked, how long was it in the air? Draw a sketch. Hypothesis: about 2 seconds. 1 Walker, Physics, page 106.

20 Time of Flight Example, #32 A soccer ball is kicked with a speed of 9.50 m/s at an angle of 25.0 above the horizontal. If the ball lands at the same level from which it was kicked, how long was it in the air? Draw a sketch. Hypothesis: about 2 seconds. Given: v 0 = 9.50 m/s, θ = 25.0 Asked for: time of flight, t 1 Walker, Physics, page 106.

21 Time of Flight Example, #32

22 such as 75 and 15. Of course, the maximum height and time of flight for one of these values of u i are different from the maximum height and time of flight for the complementary value. Time of Flight of a Projectile Quick Quiz Rank the launch angles for the five paths in the figure with respect to time of flight from the shortest time of flight respect to time of flight from the shortest time of flight to the longest. to the longest. (Assume the magnitude v i remains the same.) Q uick Quiz 4.3 Rank the launch angles for the five paths in Figure 4.10 with y (m) A 15, 30, 45, 60, 75 B 45, 30, 60, 15, 75 C 15, 75, 30, 60, 45 D 75, 60, 45, 30, Page 86, Serway & Jewett 15 v i 50 m/s Complementary values of the initial angle u i result in the same value of R x (m)

23 O FIGURE 4 12 Conceptual Exercise 5 Height and initial speed conceptual question # 6 6. Three projectiles (A, B, and C) are launched with different initial Three projectiles speeds so (A, that they B, and reach C) the are same launched maximum height, with different as shown initial speeds soin that Figure they reach List the the projectiles same maximum in order of increasing height, (a) as initial speed and (b) time of flight. Indicate a tie with an equal sign. shown. List the projectiles in order of increasing time of flight. y level from Figure 4-14 speed of the other two p (A) A, B, C (B) C, B, A (C) B, C, A (D) all the same 1 Walker, Physics, page 100. A B C O FIGURE 4 13 Conceptual Exercise 6 x 11. Child 1 thro child 2 thro acceleration less than th

24 Summary more about the height of a projectile time of flight of a projectile Homework Walker Physics: PREV: Ch 4, onward from page 100. Problems: 27, 29, 31, 33, 57, 61, 63, 67, 84 Ch 4, onward from page 100. Problems: 39, 40 & 41, 43, 87