Introduction to Mechanics Time of Flight Range of a Projectile Trajectory Equation

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1 Introduction to Mechanics Time of Flight Range of a Projectile Trajectory Equation Lana Sheridan De Anza College Feb 12, 2018

2 Last time projectiles launched horizontally projectiles launched at an angle

3 Overview time-of-flight of a projectile range of a projectile maximizing the range of a projectile trajectory equation

4 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

5 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?

6 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

7 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.

8 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.

9 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.

10 Time of Flight Example, #32 Given: v 0 = 9.50 m/s, θ = 25.0 Asked for: time of flight, t Strategy: use t flight = 2v 0 sin θ g ( ball lands at the same level from which it was kicked )

11 Time of Flight Example, #32 Given: v 0 = 9.50 m/s, θ = 25.0 Asked for: time of flight, t Strategy: use t flight = 2v 0 sin θ g ( ball lands at the same level from which it was kicked ) t flight = 2(9.50 m/s) sin(25.0 ) 9.8 m/s 2 = s Reasonable?: Less than half of my guess, but the angle it is kicked at is quite shallow, so the answer makes sense.

12 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)

13 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, 15 1 Page 86, Serway & Jewett v i 50 m/s Complementary values of the initial angle u i result in the same value of R x (m)

14 O FIGURE 4 12 Conceptual Exercise 5 Height and initial speed conceptual question 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 A B C O FIGURE 4 13 Conceptual Exercise 6 (D) all the same 1 Walker, Physics, page 106, prob 28. x 11. Child 1 thro child 2 thro acceleration less than th

15 O FIGURE 4 12 Conceptual Exercise 5 Height and initial speed conceptual question 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 A B C O FIGURE 4 13 Conceptual Exercise 6 1 Walker, Physics, page 106, prob 28. x 11. Child 1 thro child 2 thro acceleration less than th

16 Range of a Projectile range The distance in the horizontal direction that a projectile covers ion are before completely hitting the ground. time t as the com- How can we find the range of a projectile? arabolic path ty and accelera- ) nowhere (b) the at what point are to each other? y S vi v y 0 tile l case of projectile the origin at t i 5 rns to the same hori- O u i h R Figure 4.9 A projectile launched x

17 Range of a Projectile e s motion are completely ely, with time t as the coms in its parabolic path e velocity and accelerather? (a) nowhere (b) the choices, at what point are parallel to each other? y S vi v y 0 O ui h R x Projectile special case of projectile ed from the origin at t i 5 nd returns to the same hori- Figure 4.9 A projectile launched seballs, over a flat surface from the origin There footballs, is no and acceleration golf in the x-direction! (a at t i 5 0 with an initial velocity x = 0) aunched. S analyze: the peak point, vi. The maximum height of the projectile x is h, = and vthe horizontal t, which has coordinates x t range is R. At, the peak of the projectile, and the distance trajectory, the particle has coordi- is the ly in terms We just of v i, need u i, and t. g. But tnates (R/2, time h). of flight!

18 Range of a Projectile ely, with time t as the coms in its parabolic path e velocity and accelerather? (a) nowhere (b) the choices, at what point are parallel to each other? y S vi v y 0 Projectile special case of projectile ed from the origin at t i 5 nd returns to the same horiseballs, footballs, and golf aunched. analyze: the peak point, t, which has coordinates projectile, and the distance ly in terms of v i, u i, and g. O ui h R x Figure 4.9 A projectile launched over a flat x surface = vfrom x t the origin at t i 5 0 with an initial ( velocity ) S vi. The maximum height 2v0 sin of the θ R projectile = v i is cos h, and θ the horizontal range is R. At, the peak gof the trajectory, the particle has coordinates = 2v (R/2, R 0 2 sin θ cos θ h). g R = v 0 2 sin(2θ) g

19 Range of a Projectile A long jumper leaves the ground at an angle of 20.0 above the horizontal and at a speed of 11.0 m/s. How far does he jump in the horizontal direction?

20 Range of a Projectile A long jumper leaves the ground at an angle of 20.0 above the horizontal and at a speed of 11.0 m/s. How far does he jump in the horizontal direction? Draw a Sketch. Hypothesis: he ll be in the air for less than a second. Less than 11 m, more than 2 m.

21 Range of a Projectile A long jumper leaves the ground at an angle of 20.0 above the horizontal and at a speed of 11.0 m/s. How far does he jump in the horizontal direction? Draw a Sketch. Hypothesis: he ll be in the air for less than a second. Less than 11 m, more than 2 m. R = v 2 0 sin(2θ) g = (11.0 m/s)2 sin( ) 9.81 m/s 2 = 7.93 m Reasonable?: Yes, the answer is in the range I predicted.

22 Range of a Projectile ely, with time t as the coms in its parabolic path e velocity and accelerather? (a) nowhere (b) the choices, at what point are parallel to each other? y S vi v y 0 Projectile special case of projectile ed from the origin at t i 5 nd returns to the same horiseballs, footballs, and golf aunched. analyze: the peak point, t, which has coordinates projectile, and the distance ly in terms of v i, u i, and g. O ui h R x Figure 4.9 A projectile launched over a flat x surface = vfrom x t the origin at t i 5 0 with an initial ( velocity ) S vi. The maximum height 2v0 sin of the θ R projectile = v i is cos h, and θ the horizontal range is R. At, the peak gof the trajectory, the particle has coordinates = 2v (R/2, R 0 2 sin θ cos θ h). g R = v 0 2 sin(2θ) g

23 Maximizing Range R = v 2 0 sin(2θ) g What angle maximizes the range of the projectile?

Maximizing Range ATICS R = v 2 0 sin(2θ) g What angle maximizes the range of the projectile? Height, y (m) 15 12.5 10 7.5 5 = 60 = 30 = 45 2.

24 Maximizing Range ATICS R = v 2 0 sin(2θ) g What angle maximizes the range of the projectile? Height, y (m) = 60 = 30 = degrees) O Distance, x (m) (b) e in the absence R is maximized of air resistance when sin(2θ) = 1 θ = 45 or a projectile launched with an initial speed of 20 m/s. Note that the maximum range y greater than or less than 45, such as 30 and 60, give the same range. (b) Trajectories of 30 40

25 Projectile Trajectory Suppose we want to know the height of a projectile (relative to its launch point) in terms of its x coordinate. Suppose it is launched at an angle θ above the horizontal, with initial velocity v 0.

26 Projectile Trajectory Suppose we want to know the height of a projectile (relative to its launch point) in terms of its x coordinate. Suppose it is launched at an angle θ above the horizontal, with initial velocity v 0. For the x-direction: x = v 0 cos θt t = x v 0 cos θ

27 Projectile Trajectory Suppose we want to know the height of a projectile (relative to its launch point) in terms of its x coordinate. Suppose it is launched at an angle θ above the horizontal, with initial velocity v 0. For the x-direction: y-direction: Substituting for t gives: x = v 0 cos θt t = y = v 0 sin θt 1 2 gt2 x v 0 cos θ y = (tan θ)x g 2v 2 0 cos2 θ x 2

28 Projectile Trajectory ( y = (tan θ)x g 2v 2 0 cos2 θ ) x 2 This is why projectiles trace out a parabola with respect to horizontal position, as well as with respect to time.

Shepard sim Golf on the Moon with him o The projectile s path without air resistance is a symmetricalfra Mauro parabola. His distanc sand trap. With air resistance, this is no longer the case.

29 Shepard sim Golf on the Moon with him o The projectile s path without air resistance is a symmetricalfra Mauro parabola. His distanc sand trap. With air resistance, this is no longer the case. Now, w that R varie y 3.5 = 45 when sin 2u range. Thus = 35 What Happens with Air Resistance? O = 25 = 15 x Range (m) FIGURE 4 9 Projectiles with air 1 Figure fromresistance Walker, Physics, page 97. As expecte depend str 2 tional to v 0 Note th at the same

30 Validity of Range Equation Expression for the range of a projectile: When is it valid? R = v 2 0 sin(2θ) g

31 Validity of Range Equation Expression for the range of a projectile: R = v 2 0 sin(2θ) g When is it valid? when the projectile lands at the same height it is launched from! when there is no air resistance (we will not deal with air resistance in this course)

32 Summary time of flight range of a projectile trajectory equation Homework Walker Physics: PREVIOUS: Ch 4, onward from page 100. Problems: 51, 52, 77, 83 (changed!) NEW: Ch 4, onward from page 100. Conc. Ques. 7; Problems: 37, 47, 49, 83

Introduction to Mechanics Range of a Projectile Trajectory Equation More Examples

Introduction to Mechanics Range of a Projectile Trajectory Equation More Examples Lana Sheridan De Anza College Oct 25, 2017 Last time max height of a projectile time-of-flight of a projectile range of