Applying Newton’s Second Law

1. Applying Newton’s Second Law Force units: 1 N = 1 kg m/s2.

3. Forces are important in our world Forces change motion F = ma Simplest Force is F = constant Implies a = constant Constant acceleration

4. Velocity and distance Δx=vavt • In general vav is not vinst • Simple if v(t) = v • Graphical interpretation • What if v(t)? v(t) Δx=vt= area t

5. Even if v(t) • Δx = area in v/t plot

6. Examples v(t) and Δx(t)(=d(t))

9. Question The graph at right represents the velocity of a car over time. The displacement of the car can be found by A) adding the slopes of each section of the graph. B) adding the area 1 to area 2. C) subtracting area 2 from area 1. D) subtracting area 1 from area 2.

10. Question On Friday, Adam heads home for the weekend. He drives for the first two hours at a speed of 105 km/hr. The last hour, he drives at a speed of 85 km/hr. His average speed for the trip was a) less than 95 km/hr. b) equal to 95 km/hr. c) greater than 95 km/hr.

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

12. If a = const. Not true in general If a = const. one dimensional motion

13. Fig. 04.01 Vav gives Same area Hence same distance

14. Constant acceleration Δx= vavt = vit+1/2at2 ti = 0

15. Fig. 04.02

16. a=constant • 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

17. 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

18. Motion diagram X O Xv0 = 1 a = 0 O v0= 1 a = 1 X O X O

19. §4.2 Visualizing Motion with Constant Acceleration Motion diagrams for three carts:

20. a=0 v =1m/s x=vit+ ½ at2 a=0.2 v=1m/s a=-0.2 v=2m/s v=vi +at

21. Force causes a: F = ma A train of m = 55,200 kg is traveling along a straight, level track at 26.8 m/s. Suddenly the engineer sees a truck stalled on the tracks 184 m ahead. If the maximum braking force has magnitude 84 kN, can the train be stopped in time?

23. 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

24. y x w Free Fall A stone is dropped from the edge of a cliff; if air resistance can be ignored, the FBD for the stone is: Apply Newton’s Second Law The stone is in free fall; only the force of gravity acts on the stone.

25. 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?

26. 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

27. 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

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

29. Also an how far question • (vf2-vi2) =2ad • What is vf? vi? What is a? (be careful of signs!)

30. 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?

31. y x 369 m Example (text problem 4.24): 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? Given: viy = 0 m/s, ay =  9.8 m/s2, y = 369 m ay Unknown: vyf

32. Projectile Motion What happens if an object falls (vertically) at the same time it is moving with a constant horizontal velocity? The resulting curved path is a combination of horizontal and vertical motion. Horizontal component of v is completely independent of vertical component. It is not effected by (vertical) force of gravity.

35. Projectile motion • Each component is independent of the other. • Their combined effects produce the curved trajectories of projectiles.

37. Fig. 04.25

39. Question Annette throws a snowball horizontally from the top of a roof; Boris throws a snowball straight down. Once in flight, the acceleration of snowball B is ________ the acceleration of snowball A. A) greater than B) less than C) equal to

40. Question Annette and Boris release their snowballs from the same height and at the same time. Annette's is dropped while Boris' is thrown horizontally. Which one hits the ground first? A) the "dropped" snowball B) the "thrown" snowball C) they hit at the same time D) it depends on how hard Boris threw E) it depends on the initial height

41. Projectiles launched at an angle • Each component evolves independently. • y (vertical )component now has an initial value • vy = vy - gt • “Free fall” in y

43. Fig. 04.24

44. Fig. 04.27

45. Fig. 04.24

46. Fig. 04.27

47. Fig. 04.39

48. Example: An object is projected from the origin. The initial velocity components are vix = 7.07 m/s, and viy = 7.07 m/s. Determine the x and y position of the object at 0.2 second intervals for 1.4 seconds. Also plot the results. Since the object starts from the origin, y and x will represent the location of the object at time t.

49. Example continued:

50. Example continued: This is a plot of the x position (black points) and y position (red points) of the object as a function of time.