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Chapter 2 Motion Along a Straight Line One dimensional motion

Chapter 2 Motion Along a Straight Line One dimensional motion. Assignment 1. Chp 1 1-33,1-52 Chp 2 2-39, 2-72 Chp 3 3-22, 3-84 Due Monday 12:00 Mar 7. First Week Tuesday Mar 1 Ch. 1 Measurement and Estimating Wednesday Mar 2 Ch. 2 Projectile Motion Thursday Mar 3

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Chapter 2 Motion Along a Straight Line One dimensional motion

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  1. Chapter 2Motion Along a Straight LineOne dimensional motion

  2. Assignment 1 • Chp 1 1-33,1-52 • Chp 2 2-39, 2-72 • Chp 3 3-22, 3-84 Due Monday 12:00 Mar 7

  3. First Week Tuesday Mar 1 Ch. 1 Measurement and Estimating Wednesday Mar 2 Ch. 2 Projectile Motion Thursday Mar 3 Ch 2 Questions Ch. 3 Kinematics in 2D Friday Mar 4 Ch. 3 Kinematics in 2D Second Week Monday Mar 7 Ch. 4 Newton’s Laws of Motion Tuesday Mar 8 Ch 4 Newton’s Laws of Motion Wednesday Mar 9 Ch 5 Fricton, Drag forces, and Circular motion Thursday Mar 10 Ch 5 Fricton, Drag forces, and Circular motion Friday Mar 11 Ch 6 Gravitation Syllabus

  4. Third Week Monday Mar 14 Quiz 1 Ch 6 Gravitation Tuesday Mar 15 Ch. 7 Work and Energy Wednesday Mar 16 Ch. 7 Work and Energy Ch8 Conservation of Energy Thursday Mar 17 Ch 8 Conservation of Energy Friday Mar 18 Ch. 9 Linear Momentum Fourth Week Monday Mar 21 Ch. 9 Linear Momentum Tuesday Mar 22 Quiz 2 Ch. 9 Linear Momentum Wednesday Mar 23 Ch 10 Rotational Motion Thursday Mar 24 Ch 10 Rotational Motion Friday Mar 25 Ch 11 Angular Momentum Syllabus

  5. Fifth Week Monday Mar 14 Ch 11 Angular Momentum Ch 12 Static Equilibrium Tuesday Mar 15 Ch. 12 Static Equilibrium Wednesday Mar 16 Review for Final Thursday Mar 17 Final Friday Syllabus

  6. TYPICAL SPEEDS Motion v(mph) v(m/s) v/c Light 669,600,000 300,000,000 1 Earth around sun 66,600 29,600 10-4 Moon around Earth 2300 1000 3*10-6 Jet fighter 2200 980 3*10-6 Sound in air 750 334 10-6 Commercial airliner 600 267 10-6 Cheetah 62 28 10-7 Falcon diving 82 37 10-7 Olympic 100m dash 22 10 3*10-8 Flying bee 12 5 10-8 Walking ant 0.03 0.01 3*10-11 Swimming sperm 0.0001 0.000045 10-13 Nonrelativistic speeds

  7. Average velocity velocity = (distance traveled)/time A person running along a straight line at some velocity. x1 =10 m x2=20 m t1= 2 s t2= 4 s -30 -20 -10 0 10 20 30 x(m)

  8. Δx Δt Distance-time graph for running in a straight line Distance(m) Average = Instantaneous Velocity or Exact v = 5 m/s 20 10 0 6 2 4 Time(s)

  9. Constant velocity-time graph Velocity (m/s) 5 4 3 2 1 0 1 2 3 Time (s)

  10. What is meant by vavg Suppose I run for 5 s at a velocity of 2 m/s, then I rest for 5 s, and then I run for 10 s at a velocity of 2 m/s. What is my average velocity over the 20 s?

  11. x(m) 30 20 10 5 10 15 t (secs) 20 Distance-time graph for changing VWhat is meant by vavg vavg= total displacement/total time Total displacement = 10 + 0 + (30-10) = 30 m Total time = 5 + (10 - 5) + (20 -10) = 20 s = 30 / 20 = 1.50 m/s Wrong Way vav = (2m/s + 0m/s +2 m/s)/3 = 1.33 m/s 0

  12. Distance-time graph for changing V x(m) vavg = total displacement/total time = 30 / 20 = 1.50 m/s 30 20 vavg = 1.50 m/s 10 0 t(secs) 5 10 15 20

  13. What is the difference between average velocity and average speed? Suppose I run for 5 s at a velocity of 2 m/s, then I rest for 5 s, and then I run for 10 s at a velocity of 2 m/s. Now I run for 20 s at - 2m/s or backwards. What is my average velocity and speed over the entire 40 s?

  14. x(m) 30 20 10 5 10 15 20 25 30 35 40 t(secs) Average velocity and average speed are not always the same. Average velocity = vavg = total displacement/total time = (30 - 30) / 40 = 0 m/s Average speed = savg = (30 + 30)/40 = 1.5 m/s 0

  15. Here is the velocity-time graph for uniform acceleration. v(m/s) Δv 2 Δt 1 0 t (secs) Units of a are (m/s2 in mks system of units)

  16. How far does an object move from point 1 to point 2?It is equal to the total area under the green line in between points 1 and 2. v(m/s) Area =1/2 base x height + length x width 0 1 2 t (secs)

  17. v0 + at NON-ZERO INITIAL SPEED v v0 0 t

  18. Summary of Equations in 1D(constant acceleration) Under what conditions do these apply?

  19. Same Equationswith initial velocity = 0 Lets look at a numerical example and then a demo.

  20. Galileo’s Result (1564) Dropping things from rest Galileo’s experiments produced a surprising Result. All objects fall with the same acceleration Regardless of mass and shape. g = 9.8 m/s2 or 32 ft/s2 Neglecting air resistance.

  21. 10 m Free Fall Example Find the time it takes for a free-fall drop from 10 m height. Take the downward direction as positive displacement. Use two methods. Method 1 Method 2 Find

  22. 10 m Free Fall Example Find the time it takes for a free-fall drop from 10 m height. Take the downward direction as positive displacement. Use two methods. Method 1 Find

  23. Find the time it takes for a free-fall drop from 10 m height. Take the downward direction as positive displacement. Use two methods. Method 2

  24. Demos (Motion in one dimension) • Find the time between hits of free fall acceleration of three weights equally spaced on a string 50 cm apart. • The times of the first three are: The time between hits on the floor = 0.13 , 0.10, 0.09

  25. Time between hits vrs distance

  26. How do you space the weights apart such that they hit at equal successive time intervals?

  27. How far does a train go when it starts from rest and uniformly increases its speed to 120 m/s in 1 min?

  28. v = at v Find t How far does a train go when it starts from rest and uniformly increases its speed to v m/s in time t?

  29. 10 m Non-zero Initial Velocity Example Find the time t it takes for a platform diver 10 m high to hit the water if he takes off vertically with a speed of - 4 m/s and the speed v with which the diver strikes the water. Choose positive down

  30. ConcepTest 2.1 Walking the Dog You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement? 1) yes 2) no

  31. ConcepTest 2.1 Walking the Dog You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement? 1) yes 2) no Yes, you have the same displacement. Since you and your dog had the same initial position and the same final position, then you have (by definition) the same displacement. Follow-up: Have you and your dog traveled the same distance?

  32. ConcepTest 2.6b Cruising Along II 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip?

  33. ConcepTest 2.6b Cruising Along II 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip? It is not 40 mi/hr! Remember that the average speed is distance/time. Since it takes longer to cover 4 miles at the slower speed, you are actually moving at 30 mi/hr for a longer period of time! Therefore, your average speed is closer to 30 mi/hr than it is to 50 mi/hr. Follow-up: How much further would you have to drive at 50 mi/hr in order to get back your average speed of 40 mi/hr?

  34. ConcepTest 2.8a Acceleration I 1) Yes 2) No 3) Depends on the velocity If the velocity of a car is non-zero (v ≠0), can the acceleration of the car be zero?

  35. ConcepTest 2.8a Acceleration I 1) Yes 2) No 3) Depends on the velocity If the velocity of a car is non-zero (v≠0), can the acceleration of the car be zero? Sure it can! An object moving with constantvelocity has a non-zero velocity, but it has zeroacceleration since the velocity is not changing.

  36. ConcepTest 2.8b Acceleration II 1) both v = 0 and a = 0 2) v ¹ 0, but a = 0 3) v = 0, but a ¹ 0 4) both v ¹ 0 and a ¹ 0 5) not really sure When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path?

  37. ConcepTest 2.8b Acceleration II y 1) both v = 0 and a = 0 2) v 0, but a = 0 3) v = 0, but a 0 4) both v 0 and a 0 5) not really sure ¹ When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? ¹ ¹ ¹ At the top, clearly v = 0 because the ball has momentarily stopped. But the velocity of the ball is changing, so its acceleration is definitely not zero! Otherwise it would remain at rest!! Follow-up: …and the value of a is…?

  38. ConcepTest 2.11 Two Balls in the Air 1) at height h 2) above height h/2 3) at height h/2 4) below height h/2 but above 0 5) at height 0 A ball is thrown straight upward with some initial speed. When it reaches the top of its flight (at a height h), a second ball is thrown straight upward with the same initial speed. Where will the balls cross paths?

  39. ConcepTest 2.11 Two Balls in the Air 1) at height h 2) above height h/2 3) at height h/2 4) below height h/2 but above 0 5) at height 0 A ball is thrown straight upward with some initial speed. When it reaches the top of its flight (at a height h), a second ball is thrown straight upward with the same initial speed. Where will the balls cross paths? The first ball starts at the top with no initial speed. The second ball starts at the bottom with a large initial speed. Since the balls travel the same time until they meet, the second ball will cover more distance in that time, which will carry it over the halfway point before the first ball can reach it. Follow-up: How could you calculate where they meet?

  40. Differential Calculus Definition of Velocity when it is smoothly changing Define the instantaneous velocity Recall (average) as Δt 0 = dx/dt (instantaneous) Example

  41. v = Δx /Δt dx/dt = lim Δx /Δt as Δt 0 x + Δx = f(t + Δt) . x, t x = f(t) (t+Δt) t DISTANCE-TIME GRAPH FOR UNIFORM ACCELERATION x t

  42. dx/dt = lim Δx /Δt as Δt 0 Differential Calculus: an example of a derivative velocity in the x direction

  43. Problem 4-7 The position of an electron is given by the following displacement vector , where t is in secs and r is in m. What is the electron’s velocity vector v(t)? What is the electron’s velocity vector and components at t= 2 s? What is the magnitude of the velocity or speed? What is the angle relative to the positive direction of the x axis?

  44. +vy 3 +vx φ -16 What is the angle relative to the positive direction of the x axis?

  45. v=dx/dt v= at t Integral CalculusHow far does it go? tf vi 0 Δti Distance equals area under speed graph regardless of its shape Area = x = 1/2(base)(height) = 1/2(t)(at) = 1/2at2

  46. Integration:anti-derivative

  47. Three Important Rules of Differentiation Power Rule Product Rule Chain Rule

  48. Some Derivatives

  49. Some integrals

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