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Understanding One-Dimensional Motion in Physics

Dive into the world of kinematics in physics with a focus on motion along a straight line using the particle model. Learn about position-time graphs, displacement, vectors, scalars, average velocity, and instantaneous velocity. Explore concepts such as constant velocity, acceleration, and the relationship between velocity and acceleration.

wendyburns
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Understanding One-Dimensional Motion in Physics

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  1. Chapter 2 Motion in One Dimension

  2. Kinematics • Describes motion while ignoring the agents that caused the motion • For now, will consider motion in one dimension • Along a straight line • Will use the particle model • A particle is a point-like object, has mass but infinitesimal size

  3. Position • Defined in terms of a frame of reference • One dimensional, so generally the x- or y-axis • The object’s position is its location with respect to the frame of reference

  4. Position-Time Graph • The position-time graph shows the motion of the particle (car) • The smooth curve is a guess as to what happened between the data points

  5. Active Figure • AF_0201 position versus time.swf

  6. Displacement • Defined as the change in position during some time interval • Represented as x • SI units are meters (m) x can be positive or negative • Different than distance – the length of a path followed by a particle

  7. Vectors and Scalars • Vector quantities need both magnitude (size or numerical value) and direction to completely describe them • Will use + and – signs to indicate vector directions • Scalar quantities are completely described by magnitude only

  8. Average Velocity • The average velocity is rate at which the displacement occurs • The dimensions are length / time [L/T] • The SI units are m/s • Is also the slope of the line in the position – time graph

  9. Average Velocity, cont • Gives no details about the motion • Gives the result of the motion • It can be positive or negative • It depends on the sign of the displacement • It can be interpreted graphically • It will be the slope of the position-time graph

  10. Displacement from a Velocity - Time Graph • The displacement of a particle during the time interval ti to tf is equal to the area under the curve between the initial and final points on a velocity-time curve

  11. Instantaneous Velocity • The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero • The instantaneous velocity indicates what is happening at every point of time

  12. Instantaneous Velocity, graph • The instantaneous velocity is the slope of the line tangent to the x vs. t curve • This would be the green line • The blue lines show that as t gets smaller, they approach the green line

  13. Active Figure • AF_0202 velocity and slope.swf

  14. Instantaneous Velocity, equations • The general equation for instantaneous velocity is • When “velocity” is used in the text, it will mean the instantaneous velocity

  15. Instantaneous Velocity, Signs • At point A, vx is positive • The slope is positive • At point B, vx is zero • The slope is zero • At point C, vx is negative • The slope is negative

  16. Instantaneous Speed • The instantaneous speed is the magnitude of the instantaneous velocity vector • Speed can never be negative

  17. Constant Velocity • If the velocity of a particle is constant • Its instantaneous velocity at any instant is the same as the average velocity over a given time period • vx = v x avg = Dx / Dt • Also, xf = xi + vx t • These equations can be applied to particles or objects that can be modeled as a particle moving under constant velocity

  18. Average Acceleration • Acceleration is the rate of change of the velocity • It is a measure of how rapidly the velocity is changing • Dimensions are L/T2 • SI units are m/s2

  19. Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as t approaches 0

  20. Instantaneous Acceleration – Graph • The slope of the velocity vs. time graph is the acceleration • Positive values correspond to where velocity in the positive x direction is increasing • The acceleration is negative when the velocity is in the positive x direction and is decreasing

  21. Negative Acceleration • Negative acceleration does not necessarily mean that an object is slowing down • If the velocity is negative and the acceleration is negative, the object is speeding up • Deceleration is commonly used to indicate an object is slowing down, but will not be used in this text

  22. Acceleration and Velocity, 1 • When an object’s velocity and acceleration are in the same direction, the object is speeding up in that direction • When an object’s velocity and acceleration are in the opposite direction, the object is slowing down

  23. Acceleration and Velocity, 2 • The car is moving with constant positive velocity (shown by red arrows maintaining the same size) • Acceleration equals zero

  24. Acceleration and Velocity, 3 • Velocity and acceleration are in the same direction • Acceleration is uniform (blue arrows maintain the same length) • Velocity is increasing (red arrows are getting longer) • This shows positive acceleration and positive velocity

  25. Acceleration and Velocity, 4 • Acceleration and velocity are in opposite directions • Acceleration is uniform (blue arrows maintain the same length) • Velocity is decreasing (red arrows are getting shorter) • Positive velocity and negative acceleration

  26. Active Figure • AF_0208 match the motion.swf • AF_0212 acceleration and slope.swf • AF_0211 motion graphs for an accelerating car.swf

  27. Kinematic Equations – Summary

  28. Kinematic Equations • The kinematic equations may be used to solve any problem involving one-dimensional motion with a constant acceleration • You may need to use two of the equations to solve one problem • The equations are useful when you can model a situation as a particle under constant acceleration

  29. Kinematic Equations, specific • For constant a, • Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration • Does not give any information about displacement

  30. Kinematic Equations, specific • For constant acceleration, • The average velocity can be expressed as the arithmetic mean of the initial and final velocities • Only valid when acceleration is constant

  31. Kinematic Equations, specific • For constant acceleration, • This gives you the position of the particle in terms of time and velocities • Doesn’t give you the acceleration

  32. Kinematic Equations, specific • For constant acceleration, • Gives final position in terms of velocity and acceleration • Doesn’t tell you about final velocity

  33. Kinematic Equations, specific • For constant a, • Gives final velocity in terms of acceleration and displacement • Does not give any information about the time

  34. Graphical Look at Motion – displacement – time curve • The slope of the curve is the velocity • The curved line indicates the velocity is changing • Therefore, there is an acceleration

  35. Graphical Look at Motion – velocity – time curve • The slope gives the acceleration • The straight line indicates a constant acceleration

  36. Graphical Look at Motion – acceleration – time curve • The zero slope indicates a constant acceleration

  37. Problem-Solving Strategy –Constant Acceleration • Conceptualize • Establish a mental representation • Categorize • Confirm there is a particle or an object that can be modeled as a particle • Confirm it is moving with a constant acceleration

  38. Problem-Solving Strategy –Constant Acceleration, cont • Analyze • Set up the mathematical representation • Choose the instant to call the initial time and another to be the final time • These are defined by the parts of the problem of interest, not necessarily the actual beginning and ending of the motion • Choose the appropriate kinematic equation(s)

  39. Problem-Solving Strategy –Constant Acceleration, Final • Finalize • Check to see if the answers are consistent with the mental and pictorial representations • Check to see if your results are realistic

  40. Freely Falling Objects • A freely falling object is any object moving freely under the influence of gravity alone • It does not depend upon the initial motion of the object • Dropped – released from rest • Thrown downward • Thrown upward

  41. Acceleration of Freely Falling Object • The acceleration of an object in free fall is directed downward, regardless of the initial motion • The magnitude of free fall acceleration is g = 9.80 m/s2 • g decreases with increasing altitude • g varies with latitude • 9.80 m/s2 is the average at the earth’s surface

  42. Acceleration of Free Fall, cont. • We will neglect air resistance • Free fall motion is constantly accelerated motion in one dimension • Let upward be positive • Use the kinematic equations with ay = g = -9.80 m/s2 • The negative sign indicates the direction of the acceleration is downward

  43. Free Fall Example • Initial velocity at A is upward (+) and acceleration is g (-9.8 m/s2) • At B, the velocity is 0 and the acceleration is g (-9.8 m/s2) • At C, the velocity has the same magnitude as at A, but is in the opposite direction • The displacement is –50.0 m (it ends up 50.0 m below its starting point)

  44. Free Fall Terminal Velocity • AF_0518 terminal speed.swf

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