1. 2-1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 2 Motion

2. 2-2 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 2 Outline: Main Ideas How scientists describe motion  Defining “FAST”: speed  Using VECTORS  Acceleration  Distance, time, acceleration Constant Acceleration: Gravity Newton’s Laws  Defining inertia Universal Gravitation  An example: circular motion  Newton’s Law of Gravity

3. 2-3 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. What is motion? We say an object moves if the object changes it’s position That is…if the object is at one point in space at one instant in time, and a different point in space at another instant in time. Everything in the universe is in constant motion.

4. 2-4 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Describing the motion We know that “motion” means a change in position The next step is describe the change  SPEED is the term used to describe HOW FAST the location of any object is changing  Speed is the RATE OF CHANGE OF POSITION

5. 2-5 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. SPEED The AVERAGE speed is defined: speed = distance traveled  time spent v = d / t UNITS of speed: (length / time) The SI unit of speed is m/s Speedometer reads Instantaneous Speed The term RATE is scientific term that generally refers to a change in any quantity proportional to time

6. 2-6 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Important Note In a car, the speedometer reads INSTANTANEOUS SPEED  NOT average speed INSTANTANEOUS SPEED is a measure of HOW fast at a particular instant in time

7. 2-7 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Common Units for speed Any distance unit divided by any time unit is a valid unit for speed IN THE SI SYSTEM: meters per second (m/s) Other valid units:  feet per second (f/s)  miles per hour (mph or mi/h)  kilometers per hour (km/h

8. 2-8 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Number sense 10 m/s is about 22 miles per hour 20 m/s is about 45 miles per hour 30 m/s is about 67 miles per hour 40 m/s is about 90 miles per hour

9. 2-9 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Relationships The equation for speed can be used to find out HOW FAR  With knowledge of how fast, and how long v stands for average speed average speed is distance divided by time d stands for distance Distance is average speed multiplied by time t stands for time Time is distance divided by average speed

10. 2-10 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. SCALAR VS. VECTOR SCALAR QUANTITY A QUANTITY IN WHICH ONLY MAGNITUDE IS SPECIFIED VECTOR QUANTITY A QUANTITY IN WHICH BOTH MAGNITUDE AND DIRECTION ARE SPECIFIED

11. 2-11 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Examples FAMILIAR SCALAR QUANTITIES  TEMPERATURE (measured in Celsius, Fahrenheit, or Kelvin)  TIME (measured in seconds, minutes, or hours)  MASS (measured in kilograms or grams) FAMILIAR VECTOR QUANTITY  VELOCITY (measured in m/s or mph WITH DIRECTION SPECIFIED!!)

12. 2-12 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Vector Mathematics Since the vector quantity carries two pieces of information, it is often represented by a VECTOR  A straight line with an arrowhead The length of the line yields information about MAGNITUDE The arrow indicates the DIRECTION

13. 2-13 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

14. 2-14 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

15. 2-15 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. VECTOR SUM EXAMPLE Ordinarily, sum just means simple addition: 2 + 2 = 4 But when it comes to VECTORS, we must consider the DIRECTION A B Consider Vectors A and B With a coordinate system reference, geometry can be used to add the vectors

16. 2-16 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. VECTOR SUM EXAMPLE We must rearrange them “HEAD TO TAIL” in order to see where we end up A B B B B B C = A + B

17. 2-17 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18. 2-18 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. SO, WHAT IS “ACCELERATION” Acceleration is HOW FAST the velocity CHANGES. Acceleration is the rate of change of velocity.

19. 2-19 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Acceleration Acceleration is change in velocity time interval Velocity is defined by both SPEED and DIRECTION This means that Acceleration happens when there is a change in speed OR a change in direction OR both.

20. 2-20 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Common Acceleration Units Meters per second per second (m/s 2 ) feet per second per second (f/ s 2 ) miles per hour per hour (mph 2 )

21. 2-21 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. ACCELERATION vs VELOCITY MOST people are confused about the difference Velocity (or speed) is how fast your position is changing Acceleration is how fast your velocity is changing

22. 2-22 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. How can the velocity change? TWO WAYS— the magnitude can change (speed up or slow down) OR the direction can change (like when you drive on a curve) In either case, there is a non-zero acceleration!

23. 2-23 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Acceleration We went from 0 m/s to 27 m/s in 3s Acceleration is change in velocity divided by time This is a “VERY FAST” car!!

24. 2-24 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Notes about acceleration and velocity Acceleration is a vector Acceleration direction can be the SAME as the direction that the object is moving or in the OPPOSITE direction  Example? Slowing your car down! In this case, the direction of acceleration will be considered “negative”, since it’s effect is to make the velocity smaller Velocity is always in the same direction as the object is moving

25. 2-25 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Finding the final velocity If we know the initial velocity, acceleration and time, we can use the above equation to find the final velocity

26. 2-26 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Special case: constant acceleration If we know  Initial velocity  Final velocity  Acceleration Then we can determine HOW FAR an object travels in a given amount of time Recall that distance traveled is equal to the average velocity multiplied by the time:

27. 2-27 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Finding HOW FAR The average velocity is simply

28. 2-28 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Finding HOW FAR Then, we use this average velocity expression in our equation for d:

29. 2-29 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example You are driving your car and approaching a stop sign. You apply the brakes when your speedometer reads 27 m/s (60 mph) If you know that the breaks will provide an acceleration of a= - 6 m/s 2, how far away must you apply the brakes? The time it takes to reach the stop sign is 4.5 s or about 5.0 s.

30. 2-30 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Solution, braking distance 60 m is approximately 197 feet!

31. 2-31 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Summarize what we know so far: Speed is the total distance traveled divided by the total time Average Velocity is the change in position divided by total time Acceleration is the change in velocity divided by the time taken (two ways)  either change the magnitude of the velocity vector and/or  change the direction of the velocity vector

32. 2-32 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Now, another special case: FREE FALL Objects moving under the influence of only gravity Major contributor: GALELEO

33. 2-33 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

34. 2-34 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Galeleos Experiments: dropping the ball Actually, Galileo performed experiments by dropping a stone He found that the higher a stone is when it is dropped, the larger the speed of the stone when it hit the ground.

35. 2-35 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Galeleos Experiments: dropping the ball Actually, Galileo performed experiments by dropping a stone He found that the higher a stone is when it is dropped, the larger the speed of the stone when it hit the ground.

36. 2-36 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Galeleos Experiments: dropping the ball Actually, Galileo performed experiments by dropping a stone He found that the higher a stone is when it is dropped, the larger the speed of the stone when it hit the ground.

37. 2-37 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. GALILEO was a GENIUS! He realized that the ACCELERATION acting on the stone is CONSTANT… THIS MEANS that the velocity is DIRECTLY PROPORTIONAL to the time… The velocity of a falling object is proportional to the length of time that the object has been falling.

38. 2-38 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. What Galileo discovered... If we neglect air resistance, we see that ALL OBJECTS FALL AT THE SAME RATE The letter ‘g’ is used to represent the constant value of the acceleration due to gravity (near the surface of the Earth) g = 9.8 m/s 2 (English Units: 32 ft/s 2 ) What does this really mean?

39. 2-39 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Acceleration due to gravity… The ACCELERATION DUE TO GRAVITY is approximately 9.8 m/s 2 (near the surface of the earth) This means that gravity causes a change in the velocity of any object by 9.8 m/s EVERY SECOND So, drop any object and let it fall toward the ground:  After 1s, the speed of the object is 9.8m/s  After 2s, the speed of the object is 19.6m/s  Etc.

40. 2-40 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. HEADS UP!!! Acceleration is a VECTOR Acceleration changes VELOCITY Acceleration due to gravity ACTS ONLY IN THE VERTICAL DIRECTION

41. 2-41 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Well, what does that mean? Its easy to see if we throw a ball STRAIGHT UP Velocity is only in UP/DOWN direction (same as “g”) What happens to the velocity?

42. 2-42 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The ball goes up We have a situation to consider…The ball’s direction of travel is in the OPPOSITE direction of the acceleration, SO, the speed will DECREASE Let’s say we gave the ball an initial upward velocity of about 20 m/s After 2 s, what will the velocity of the ball be??

43. 2-43 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Back to “HOW FAR?” Recall that But for the dropped object, let d become ‘h’ (to stand for height), and v1=0 (starts from rest), with the acceleration a=g (only due to gravity:

44. 2-44 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. From this, we can answer “HOW LONG” If we know the height from which an object is dropped, we can use the above equation to determine HOW LONG the object will fall:

45. 2-45 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example: motion in one direction Hint: Remember, the distance traveled by an object that is subject to the downward acceleration of gravity is described by Lots of players brag about their long “hang time”. Some people claim to have hang times of 2 or 3 seconds... Now that you know some physics, how can you show that this cannot be true?

46. 2-46 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Let’s see how FAR (how high) you could go with a 2 second “hang time”: 2 second hang time means total time from jump to landing, so let’s say that it took the athlete half the time to go up, 1 s : d upward = (1/2) ( 9.8 m/s 2 ) x (t) 2 = 4.9 x (1) 2 = 4.9 meters NO WAY!!!

47. 2-47 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. REALITY: Even the good players get at most a jump height  2 feet (0.6 meters) this gives a “hang time” of 0.6 = (1/2) x g x (t) 2 solving this for the time, t gives t =  (0.6/4.9) = 0.35 seconds Or about 1/3 of a second…..

48. 2-48 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Path of the projectile (TRAJECTORY) PARABOLIC Trajectory HIGHEST POINT HALF WAY POINT OF HORIZONTAL DISPLACEMENT Half total time

49. 2-49 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The famous NEWTON’s LAWS LAW 1 When no net force acts on a body, a body at rest will remain at rest and a body in motion will remain in motion. ( “LAW OF INERTIA”) LAW 2 The acceleration of any object is directly proportional to the NET FORCE on the object and inversely (a = F / m) proportional to the mass of the object. LAW3 If one object exerts a force on a second object, then the second object exerts an equal but oppositely directed force on the first object (For every action there is a reaction)

50. 2-50 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. NEWTON2 “The law of motion” UNIFORM MOTION constant velocity: no acceleration (net force =0)

51. 2-51 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. What is a NEWTON?? 1 NEWTON = 1 kg x (1m/s 2 ) massacceleration F= m x a a = F / m NON-UNIFORM MOTION velocity is changing due to acceleration

52. 2-52 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example: Newton’s second law How much force, in NEWTONS, is needed to accelerate a truck with a mass of 3200 kg from a full stop to 60 miles per hour, if it takes about 10 seconds to reach a speed of 60 miles per hour. Solution To find the FORCE, we need to find the ACCELERATION Acceleration = change in velocity / time

53. 2-53 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Convert everything to metric units Truck mass: 3200 kg Truck speed: 60 miles per hour- need to go metric

54. 2-54 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Action-Reaction Name the action-reaction pairs: Foot applies force to a ball One person pulls the hands of a second person

55. 2-55 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Discussion Example Horizontal displacement (ignoring air resistance) Calculate how far a safety net should be placed from a human being, being shot from a cannon. The initial velocity is 40 m/s along the vertical direction and 2 m/s along the horizontal direction (Hint, only the vertical velocity will change)

56. 2-56 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. How long is the daredevil in the air? Vertical component of velocity is 40 m/s  this is the same as throwing the ball straight up with a velocity of 40 m/s,  it will take approximately 4 s to get to the top, and 4 s to get back down: TOTAL TIME = 8 s IN 8 s, with a horizontal velocity of 2 m/s, the guy travels distance = speed x time = 2 (m/s) x 8 (s) = 16 m

57. 2-57 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Projectile Motion NOTES The path taken by an object thrown upward at Earth's surface is a parabola. gravity can only act ONLY on the VERTICAL component HORIZONTAL component

58. 2-58 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. So far, we have been neglectful We have neglected Even though lots of things would not be possible without it: Stopping your car Driving your car at all (rolling motion)…

59. 2-59 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. And now, a word from our good pal, FRICTION In real life--frictional forces are almost always present. The direction of the friction force ALWAYS OPPOSES THE MOTION Friction is bad: it dissipates energy Friction is good: it allows us to roll, turn,...and...stop!