1. Motion in One Dimension Physics 2053 Lecture Notes 02a dx dt x t Kinematics in One Dimension (Phy 2053) vittitoe

2. 2-01 Displacement 2-02 Velocity 2-03 Acceleration 2-04 Motion Diagrams Motion in One Dimension Sections 2-05 One Dimensional Motion with Constant Acceleration 2-06 Freely Falling Objects Kinematics in One Dimension (Phy 2053) vittitoe

3. v  In the study of kinematics, we consider a moving object as a particle. A particle is a point-like mass having infinitesimal size and a finite mass. Displacement Kinematics in One Dimension (Phy 2053) vittitoe

4. 0246 22 44 66 x Displacement The displacement of a particle is defined as its change in position. (m)  x = x  x o = 6 m  2 m = 4 m Note: Displacement to the right is positive Kinematics in One Dimension (Phy 2053) vittitoe

5. 0246 22 44 66 x The displacement of a particle is defined as its change in position. (m)  x = x  x o =  6 m  6 m =  12 m Note: Displacement to the left is negative Displacement Kinematics in One Dimension (Phy 2053) vittitoe

6. 0246 22 44 66 x The displacement of a particle is defined as its change in position. (m)  x = x  x o = (  m)  (  6 m) = 8 m Note: Displacement to the right is positive Displacement Kinematics in One Dimension (Phy 2053) vittitoe

7. Displacement Kinematics in One Dimension (Phy 2053) vittitoe A student walks 70 m East, then walks 30 km West. What is the magnitude of the students net displacement? A) 30 m B) 40 m C) 70 m D) 100 m

8. Average velocity The average velocity of a particle is defined as x t x1x1 x2x2 t1t1 t2t2 xx tt Velocity is represented by the slope on a displacement-time graph Velocity Kinematics in One Dimension (Phy 2053) vittitoe

9. Average speed The average speed of a particle is defined as Velocity Kinematics in One Dimension (Phy 2053) vittitoe

10. Instantaneous velocity The instantaneous velocity v, equals the limiting value of the ratio xx tt x t Instantaneous velocity is represented by the slope of a displacement-time graph Velocity Kinematics in One Dimension (Phy 2053) vittitoe

11. Instantaneous speed The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity. Velocity Kinematics in One Dimension (Phy 2053) vittitoe

12. Average acceleration The average acceleration of a particle is defined as the change in velocity  v x divided by the time interval  t during which that change occurred. v t v1v1 v2v2 t1t1 t2t2 vv tt Acceleration is represented by the slope on a velocity-time graph Acceleration Kinematics in One Dimension (Phy 2053) vittitoe

13. Acceleration Kinematics in One Dimension (Phy 2053) vittitoe A new car manufacturer advertises that their car can go "from zero to sixty in 8 s". This is a description of A) instantaneous acceleration. B) average speed. C) instantaneous speed. D) average acceleration.

14. Instantaneous acceleration The instantaneous acceleration equals the derivative of the velocity with respect to time vv tt v t Instantaneous acceleration is represented by the slope of a velocity-time graph Acceleration Kinematics in One Dimension (Phy 2053) vittitoe

15. Acceleration Kinematics in One Dimension (Phy 2053) vittitoe A moving car experiences a constant acceleration of 1.5 m/s 2. This means the car is A) traveling at 1.5 m/s in every second. B) changing its velocity by 1.5 m/s. C) increasing its velocity by 1.5 m/s in every second. D) increases its displacement by 1.50 m each second.

16. True or False? (a) A car must always have an acceleration in the same direction as its velocity Quick Quiz 2.2 (page 32) (b) It’s possible for a slowing car to have a positive acceleration (c) An object with constant nonzero acceleration can never stop and stay stopped. Kinematics in One Dimension (Phy 2053) vittitoe

17. The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Motion Diagrams Kinematics in One Dimension (Phy 2053) vittitoe

18. The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Motion Diagrams (con’t) Kinematics in One Dimension (Phy 2053) vittitoe

19. The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Motion Diagrams (con’t) Kinematics in One Dimension (Phy 2053) vittitoe

20. The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Motion Diagrams (con’t) Kinematics in One Dimension (Phy 2053) vittitoe

21. The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals (a) 0 to 2 s (b) 0 to 4 s (c) 2 s to 4 s (d) 4 s to 7 s (e) 0 to 8 s. Motion Diagrams (con’t) Kinematics in One Dimension (Phy 2053) vittitoe

22. 1624128402028 (s) x 4 8 12 16 20 24 28 (m) 1234 t 5 Motion Diagrams An object starts from rest and moves with constant acceleration. Kinematics in One Dimension (Phy 2053) vittitoe

23. (s) x 4 8 12 16 20 24 28 (m) 1234 t 5 12345 t (s) 2 4 6 8 10 v (m/s) Displacement 25 m Motion Diagrams Kinematics in One Dimension (Phy 2053) vittitoe

24. t x t v t a Displacement, velocity and acceleration graphs The slope of a velocity-time graph represents acceleration The slope of a displacement-time graph represents velocity Motion Diagrams Kinematics in One Dimension (Phy 2053) vittitoe

25. t x t v t a tt Displacement, velocity and acceleration graphs The area under an acceleration-time graph represents change in velocity. vv The area under a velocity-time graph represents displacement. xx Motion Diagrams Kinematics in One Dimension (Phy 2053) vittitoe

26. Motion Diagrams Kinematics in One Dimension (Phy 2053) vittitoe The slope of a position versus time graph gives A) position. B) velocity. C) acceleration. D) displacement.

27. Motion Diagrams Kinematics in One Dimension (Phy 2053) vittitoe The slope of a velocity versus time graph gives A) position. B) velocity C) acceleration D) displacement

28. Definitions of velocity and acceleration Average velocity Average acceleration One Dimensional Motion with Constant Acceleration Kinematics in One Dimension (Phy 2053) vittitoe

29. For constant acceleration An object moving with an initial velocity v o undergoes a constant acceleration a for a time t. Find the final velocity. vovo time = 0time = t Solution: Eq 1 a ? One Dimensional Motion with Constant Acceleration Kinematics in One Dimension (Phy 2053) vittitoe

30. What are we calculating? 0 t a VV One Dimensional Motion with Constant Acceleration Kinematics in One Dimension (Phy 2053) vittitoe

31. One Dimensional Motion with Constant Acceleration Objects A and B both start at rest. They both accelerate at the same rate. However, object B accelerates for twice the time as object A. What is the final speed of object B compared to that of object A? A) the same speed B) twice as fast C) three times as fast D) four times as fast

32. For constant acceleration An object moving with a velocity v o is passing position x o when it undergoes a constant acceleration a for a time t. Find the object’s displacement. Solution: time = 0time = t xoxo ? a vovo Eq 2 One Dimensional Motion with Constant Acceleration Kinematics in One Dimension (Phy 2053) vittitoe

33. What are we calculating? 0 t vovo v One Dimensional Motion with Constant Acceleration at Kinematics in One Dimension (Phy 2053) vittitoe

34. One Dimensional Motion with Constant Acceleration Kinematics in One Dimension (Phy 2053) vittitoe Objects A and B both start at rest. They both accelerate at the same rate. However, object B accelerates for twice the time as object A. What is the distance traveled by object B compared to that of object A? A) the same distance B) twice as far C) three times as far D) four times as far

35. Eq 1 Eq 2 Solve Eq 1 for a and sub into Eq 2: Solve Eq 1 for t and sub into Eq 2: Eq 3 Eq 4 One Dimensional Motion with Constant Acceleration Kinematics in One Dimension (Phy 2053) vittitoe

36. One Dimensional Motion with Constant Acceleration Kinematics in One Dimension (Phy 2053) vittitoe When the velocity of an object is zero, must its acceleration also be zero? A) no, an object thrown upward will have zero velocity at its highest point. B) no, a falling object will have zero velocity after hitting the ground. C) yes, if the object is not moving it can not be accelerating. D) yes, acceleration implies a changing velocity, it can not be zero.

37. Freely Falling Objects Kinematics in One Dimension (Phy 2053) vittitoe When an object is released from rest and falls in the absence of air resistance, which of the following is true concerning its motion? A) Its acceleration is constant B) Its velocity is constant. C) Neither its acceleration nor its velocity is constant. D) Both its acceleration and its velocity are constant.

38. Problem Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s 2. (a) How long does it take for the lead car to stop? Kinematics in One Dimension (Phy 2053) vittitoe

39. Problem Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s 2. (b) How far does the lead car travel during the acceleration? Kinematics in One Dimension (Phy 2053) vittitoe

40. Alternate Solutions Problem Kinematics in One Dimension (Phy 2053) vittitoe

41. Problem Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s 2. (c) Assuming that the chasing car brakes at the same time as the lead car, what must be the chasing car’s minimum negative acceleration so as not to hit the lead car? Kinematics in One Dimension (Phy 2053) vittitoe

42. Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2. (d) How long does it take for the chasing car to stop? Problem Kinematics in One Dimension (Phy 2053) vittitoe

43. Alternate Solutions Problem Kinematics in One Dimension (Phy 2053) vittitoe

44. A Cessna aircraft has a lift-off speed of 120 km/h. (a) What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff run of 240 m? Problem Kinematics in One Dimension (Phy 2053) vittitoe

45. A Cessna aircraft has a lift-off speed of 120 km/h. (b) How long does it take the aircraft to become airborne? Problem Kinematics in One Dimension (Phy 2053) vittitoe

46. A drag racer starts her car from rest and accelerates at 10.0 m/s 2 for a distance of 400 m (1/4 mile). (a) How long did it take the race car to travel this distance? Problem Kinematics in One Dimension (Phy 2053) vittitoe

47. A drag racer starts her car from rest and accelerates at 10.0 m/s 2 for a distance of 400 m (1/4 mile). (b) What is the speed of the race car at the end of the run? Problem Kinematics in One Dimension (Phy 2053) vittitoe

48. A ball is thrown vertically upward with a speed of 25.0 m/s. (a) How high does it rise? Problem Kinematics in One Dimension (Phy 2053) vittitoe

49. A ball is thrown vertically upward with a speed of 25.0 m/s. (b) How long does it take to reach its highest point? Problem Kinematics in One Dimension (Phy 2053) vittitoe

50. A ball is thrown vertically upward with a speed of 25.0 m/s. (c) How long does the ball take to hit the ground after it reaches its highest point? Problem Kinematics in One Dimension (Phy 2053) vittitoe

51. A ball is thrown vertically upward with a speed of 25.0 m/s. (d) What is its velocity when it returns to the level from which it started? Problem Kinematics in One Dimension (Phy 2053) vittitoe

52. Average velocity Average acceleration Kinematics with Constant Acceleration Definitions Review Kinematics in One Dimension (Phy 2053) vittitoe

53. t x t v t a tt vv xx t x t v t a Review Kinematics in One Dimension (Phy 2053) vittitoe

54. Problem Solving Skills 1. Read the problem carefully 2. Sketch the problem 3. Visualize the physical situation 4. Identify the known and unknown quantities 5. Identify appropriate equations 6. Solve the equations 7. Check your answers Review Kinematics in One Dimension (Phy 2053) vittitoe

56. Suppose that an object is moving with a constant velocity. Make a statement concerning its acceleration. A) The acceleration must be constantly increasing. B) The acceleration must be constantly decreasing. C) The acceleration must be a constant non-zero value. D) The acceleration must be equal to zero. Constant Velocity

57. Can an object have increasing speed while the magnitude of its acceleration is decreasing? Support your answer with an example. A) No, this is impossible because of the way in which acceleration is defined. B) No, because if acceleration is decreasing the object will be slowing down. C) Yes, and an example would be an object falling in the absence of air friction. D) Yes, and an example would be an object released from rest in the presence of air friction. Freely Falling Objects

58. Suppose a ball is thrown straight up. Make a statement about the velocity and the acceleration when the ball reaches the highest point. A) Both its velocity and its acceleration are zero. B) Its velocity is zero and its acceleration is not zero. C) Its velocity is not zero and its acceleration is zero. D) Neither its velocity nor its acceleration is zero. Freely Falling Objects

59. Ball A is dropped from the top of a building. One second later, ball B is dropped from the same building. As time progresses, the distance between them A) increases. B) remains constant. C) decreases. D) cannot be determined from the information given. Freely Falling Objects