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  1. 1. Chapter 2 Motion in One Dimension 01/15/14 IB Physics (IC NL) 1
  2. 2. Mechanics Mechanics is a branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements also it deals with matter and investigates energy. Mechanics is divided into three branches: 1- Statics 2- Kinematics 3- Dynamics In this slide we will discuss 01/15/14 KINEMATICS. IB Physics (IC NL) 2
  3. 3. Kinematics  Kinematics is a part of mechanics that studies motion in relationship to time.   In kinematics, you are interested in the description of motion Not concerned with the cause of the motion (Dynamics) or bodies at rest (statics) 01/15/14 IB Physics (IC NL) 3
  4. 4. Motion    A body or object is said to be in motion when it is observed moving or changing position relative to a fixed reference as time passes. Therefore motion is relative; example when you are in a car and you take the moving car as your reference then you see a building moving back relative to you. Then there should exist a reference frame to your work and since we are studying this topic while on Earth, we suppose the Earth is our reference frame thus it is called the terrestrial frame. 01/15/14 IB Physics (IC NL) 4
  5. 5. Motion cont.   Motion, to our level, is studied either according to trajectory or according to speed. The trajectory is the path followed by the center of mass of a moving object. Remark: our object could be: 1- either a solid body of considerable dimensions with a center of mass where all the mass of the object is concentrated (also called center of gravity) 2- or a point mass with relative negligible dimensions and the mass is concentrated at the point itself. Hint: don’t worry about this the question usually identifies the type of the object.  01/15/14 IB Physics (IC NL) 5
  6. 6. Motion according to trajectory. Studying the motion according to trajectory is very wide so we are limiting it to a general one which is curvilinear and includes one of three:  Rectilinear: along a straight line  Circular: along a circle  Curvilinear along a curve. The first study is the rectilinear motion that is motion in one dimension.  01/15/14 IB Physics (IC NL) 6
  7. 7. Quantities in Motion  Any motion involves three concepts     Displacement Velocity Acceleration These concepts can be used to study objects in motion 01/15/14 IB Physics (IC NL) 7
  8. 8. Position  Defined in terms of a frame of reference   One dimensional, so generally the xor y-axis Defines a starting point for the motion 01/15/14 IB Physics (IC NL) 8
  9. 9. Displacement  Defined as the change in position or how much the object was displaced from its initial position.  ∆x ≡ xf − xi    01/15/14 f stands for final and i stands for initial May be represented as ∆y if vertical Units are meters (m) in SI, centimeters (cm) in cgs. IB Physics (IC NL) 9
  10. 10. Displacements 01/15/14 IB Physics (IC NL) 10
  11. 11. Vector and Scalar Quantities  Vector quantities need both magnitude (size) and direction to completely describe them    Generally denoted by boldfaced type and an arrow over the letter + or – sign is also used in vector representation. Scalar quantities are completely described by magnitude only 01/15/14 IB Physics (IC NL) 11
  12. 12. Displacement Isn’t Distance  The displacement of an object is not the same as the distance it travels  Example: Throw a ball straight up and then catch it at the same point you released it   01/15/14 The distance is twice the height The displacement is zero IB Physics (IC NL) 12
  13. 13. Speed  The average speed of an object is defined as the total distance traveled divided by the total time elapsed total distance Average speed = total time d v = t  01/15/14 Speed is a scalar quantity IB Physics (IC NL) 13
  14. 14. Speed, cont    Average speed totally ignores any variations in the object’s actual motion during the trip The total distance and the total time are all that is important SI unit m.s-1 01/15/14 IB Physics (IC NL) 14
  15. 15. Velocity    It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs or simply the rate of change of displacement. ∆ x xf − xi v average = = ∆t tf − ti generally use a time interval, so let ti = 0 01/15/14 IB Physics (IC NL) 15
  16. 16. Velocity continued  Direction will be the same as the direction of the displacement (time interval is always positive)   + or - is sufficient Units of velocity is also m.s-1 (SI)  01/15/14 Other units may be given in a problem, but generally will need to be converted to m.s-1 these like km/h IB Physics (IC NL) 16
  17. 17. Speed vs. Velocity   Cars on both paths have the same average velocity since they had the same displacement in the same time interval The car on the blue path will have a greater average speed since the distance it traveled is larger 01/15/14 IB Physics (IC NL) 17
  18. 18. Graphical Interpretation of Velocity    Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions An object moving with a constant velocity will have a graph that is a straight line. 01/15/14 IB Physics (IC NL) 18
  19. 19. Average Velocity, Constant    The straight line indicates constant velocity The slope of the line is the value of the average velocity Determine the equation of this straight line. x= 2t - 20 01/15/14 IB Physics (IC NL) 19
  20. 20. Average Velocity, Non Constant   The motion in this type of motion is of non-constant velocity The average velocity is the slope of the blue line joining two points 01/15/14 IB Physics (IC NL) 20
  21. 21. Instantaneous Velocity    The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero lim ∆x v ≡ ∆t → 0 ∆t The instantaneous velocity indicates what is happening at every instant of time. It is a function of time. The speedometer of a car indicates the instantaneous speed. 01/15/14 IB Physics (IC NL) 21
  22. 22. Instantaneous Velocity on a Graph  The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time  01/15/14 The instantaneous speed is defined as the magnitude of the instantaneous velocity IB Physics (IC NL) 22
  23. 23. Instantaneous Velocity on a Graph Determine the instantaneous velocity at x = 15 m. About 2.5 ms-1 01/15/14 IB Physics (IC NL) 23
  24. 24. Motion according to speed In this part we study the motion of an object depending on its velocity along a rectilinear trajectory.  Our study is limited now to: 1- motion with constant speed or velocity and called uniform motion (URM) 2- motion with a constantly varying velocity and called uniformly accelerated motion. (UARM)  01/15/14 IB Physics (IC NL) 24
  25. 25. Uniform rectilinear motion   URM is motion with constant velocity The instantaneous velocities are always the same  01/15/14 All the instantaneous velocities will also equal the average velocity IB Physics (IC NL) 25
  26. 26. Kinematic equations of URM          a=0 v = constant x = vt + xo Where: a is acceleration v is velocity x is displacement xo is initial displacement or distance covered when started timing and t is time 01/15/14 IB Physics (IC NL) 26
  27. 27. Acceleration    Changing velocity (not-uniform) means an acceleration is present Acceleration is the rate of change of the velocity ∆v vf − vi a= = ∆t tf − ti Units is ms-² (SI) 01/15/14 IB Physics (IC NL) 27
  28. 28. Relationship Between Acceleration and Velocity   Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero 01/15/14 IB Physics (IC NL) 28
  29. 29. Relationship Between Velocity and Acceleration Velocity and acceleration are in the same direction  Acceleration is uniform (blue arrows maintain the same length)  Velocity is increasing uniformly (red arrows are getting longer)  Positive velocity and positive acceleration, thus Accelerated motion. va>0 01/15/14 29 IB Physics (IC NL) 
  30. 30. Relationship Between Velocity and Acceleration     Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing uniformly (red arrows are getting shorter) Velocity is positive and acceleration is negative, thus decelerated motion. va<0 01/15/14 IB Physics (IC NL) 30
  31. 31. Kinematic Equations  Used in situations with uniform acceleration a = const. 1 v = u + at 2 1 3 ∆x = vt = ( u + v ) t 2 1 2 4 ∆x = ut + at 2 v 2 = u2 + 2a∆x 5 01/15/14 IB Physics (IC NL) 31
  32. 32. Kinematic equations cont. Equation 4 is derive when plugging equation 2 in equation 3  Equation 5 is obtained when eliminating t from equation 2 and plug it in equation 3  01/15/14 IB Physics (IC NL) 32
  33. 33. Notes on the equations v i +v f  ∆ x = v average t =  t  2    Gives displacement as a function of velocity and time. Use when you don’t know and aren’t asked for the acceleration. 01/15/14 IB Physics (IC NL) 33
  34. 34. Notes on the equations v = u + at   Shows velocity as a function of acceleration and time Use when you don’t know and aren’t asked to find the displacement 01/15/14 IB Physics (IC NL) 34
  35. 35. Graphical Interpretation of the Equation u u 01/15/14 IB Physics (IC NL) 35
  36. 36. Notes on the equations 1 2 ∆ x = ut + at 2   Gives displacement as a function of time, velocity and acceleration Use when you don’t know and aren’t asked to find the final velocity 01/15/14 IB Physics (IC NL) 36
  37. 37. Displacement-time graph 01/15/14 IB Physics (IC NL) 37
  38. 38. Acceleration graph  When you are given a displacementtime graph and the figure simulates a smiling face then the motion is accelerated. If it is a gloomy face then the motion is decelerated. 01/15/14 IB Physics (IC NL) 38
  39. 39. Notes on the equations 2 2 v = u + 2a∆ x   Gives velocity as a function of acceleration and displacement Use when you don’t know and aren’t asked for the time 01/15/14 IB Physics (IC NL) 39
  40. 40. Graphical interpretation revisited  The area under the velocity-time graph represents the displacement. Note: the area above the time axis is positive displacement where as the one below the axis Is negative. Given the following diagram: the displacement is: x = 64 m 01/15/14 IB Physics (IC NL) 40
  41. 41. Graphical interpretation revisited The area under the accelerationtime graph represents the change in velocity.  Given the following graph:  What is the velocity relative to this motion: v = 45 ms-1  01/15/14 IB Physics (IC NL) 41
  42. 42. Problem-Solving Hints   Read the problem Draw a diagram   Label all quantities, be sure all the units are consistent    Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations Convert if necessary Specify the type of motion according to a. Choose the appropriate kinematic equation 01/15/14 IB Physics (IC NL) 42
  43. 43. Problem-Solving Hints, cont  Solve for the unknowns   You may have to solve two equations for two unknowns Check your results   01/15/14 Estimate and compare Check units IB Physics (IC NL) 43
  44. 44. Galileo Galilei    1564 - 1642 Galileo formulated the laws that govern the motion of objects in free fall Also looked at:     01/15/14 Inclined planes Relative motion Thermometers Pendulum IB Physics (IC NL) 44
  45. 45. Free Fall  All objects moving under the influence of gravity only are said to be in free fall    Free fall does not depend on the object’s original motion All objects falling near the earth’s surface fall with a constant acceleration The acceleration is called the acceleration due to gravity, and indicated by g 01/15/14 IB Physics (IC NL) 45
  46. 46. Laws of free fall    First law: in vacuum and in the same place, all bodies, dense or light obey the same law of fall. Second law: the trajectory followed by a freely falling object is the vertical, directed towards the center of the earth. Third law: the motion of a freely falling object is uniformly accelerated of acceleration g. 01/15/14 46 IB Physics (IC NL)
  47. 47. Free fall on the moon 01/15/14 IB Physics (IC NL) 47
  48. 48. Hyperlink for previous page  Apollo 15 Hammer/Feather gravity demonstration 01/15/14 IB Physics (IC NL) 48
  49. 49. Motion in air    Consequence: In air and in the same place, dense bodies only and at the beginning of their fall obey the same law of fall as in vacuum. When an object falls in air it is subjected to friction with air particles and to an opposing force called air resistance R. R=kSV2 01/15/14 (go Physics the following slide) 49 IB to (IC NL)
  50. 50. Air resistance cont. R = kSV2 R is air resistance k is the aerodynamic constant depending on the shape of the object. S is cross-sectional area of object. V is velocity of object. 01/15/14 IB Physics (IC NL) 50
  51. 51. Aerodynamic shape and k. For each shape there is a different k. The last shape F has the least k so that air resistance is minimal thus this shape is recommended for the use in airplanes and it is called the aerodynamic shape. Note: the object is moving to the left. 01/15/14 IB Physics (IC NL) 51
  52. 52. Terminal velocity   When the air resistance reaches a value that balances the weight of the object then Ksv2=mg mg v= ks  This velocity is called the terminal velocity with which the object continues falling and it is constant. 01/15/14 IB Physics (IC NL) 52
  53. 53. Acceleration due to Gravity   Symbolized by g g = 9.80 ms-²   g is always directed downward   When estimating, use g ≈ 10 m.s-2 toward the center of the earth Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion 01/15/14 IB Physics (IC NL) 53
  54. 54. Free Fall – an object dropped    Initial velocity is zero Let up be positive Use the kinematic equations   u= 0 a=g Generally use y instead of x since vertical Acceleration is g = -9.80 m.s-2 01/15/14 IB Physics (IC NL) 54
  55. 55. Free Fall – an object thrown downward   a = g = -9.80 m.s-2 Initial velocity ≠ 0  01/15/14 With upward being positive, initial velocity will be negative IB Physics (IC NL) 55
  56. 56. Free Fall -- object thrown upward    Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero a = g = -9.80 m.s-2 everywhere in the motion 01/15/14 IB Physics (IC NL) u=0 56
  57. 57. Thrown upward, cont.  The motion may be symmetrical    Then tup = tdown Then v = -u The motion may not be symmetrical  You can break the motion into various parts to understand but it is recommendable that you don’t do this and stick to one system of axes with specified directions that don’t change with the motion of the object. Generally in initial direction of motion of the 01/15/14 IB Physics (IC NL) object.  57
  58. 58. Non-symmetrical Free Fall   No need to divide the motion into segments Object thrown up: Vertical axis directed upwards  Horizontal axis through projection point O.  Above O, displacement is positive  Below O, displacement is negative. 01/15/14 IB Physics (IC NL)  58
  59. 59. Combination Motions 01/15/14 IB Physics (IC NL) 59