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Kinematics and One Dimensional Motion: Non-Constant Acceleration 8.01 W01D3

Kinematics and One Dimensional Motion: Non-Constant Acceleration 8.01 W01D3. Today’s Reading Assignment: W02D1. Young and Freedman: Sections 2.1-2.6. Kinematics and One Dimensional Motion. Kinematics Vocabulary. Kinema means movement in Greek Mathematical description of motion Position

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Kinematics and One Dimensional Motion: Non-Constant Acceleration 8.01 W01D3

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  1. Kinematics and One Dimensional Motion:Non-Constant Acceleration8.01W01D3

  2. Today’s Reading Assignment: W02D1 Young and Freedman: Sections 2.1-2.6

  3. Kinematics and One Dimensional Motion

  4. Kinematics Vocabulary • Kinema means movement in Greek • Mathematical description of motion • Position • Time Interval • Displacement • Velocity; absolute value: speed • Acceleration • Averages of the latter two quantities.

  5. Coordinate System in One Dimension Used to describe the position of a point in space A coordinate system consists of: • An origin at a particular point in space • A set of coordinate axes with scales and labels • Choice of positive direction for each axis: unit vectors • Choice of type: Cartesian or Polar or Spherical Example: Cartesian One-Dimensional Coordinate System

  6. Position • A vector that points from origin to body. • Position is a function of time • In one dimension:

  7. Displacement Vector Change in position vector of the object during the time interval

  8. Concept Question: Displacement • An object goes from one point in space to another. After the object arrives at its destination, the magnitude of its displacement is: • 1) either greater than or equal to • 2) always greater than • 3) always equal to • 4) either smaller than or equal to • 5) always smaller than • 6) either smaller or larger than • the distance it traveled.

  9. Average Velocity The average velocity, , is the displacement divided by the time interval The x-component of the average velocity is given by

  10. Instantaneous Velocityand Differentiation • For each time interval , calculate the x-component of the average velocity • Take limit as sequence of the x-component average velocities • The limiting value of this sequence is x-component of the instantaneous velocity at the time t .

  11. Instantaneous Velocity x-component of the velocity is equal to the slope of the tangent line of the graph of x-component of position vs. time at time t

  12. Concept Question: One Dimensional Kinematics The graph shows the position as a function of time for two trains running on parallel tracks. For times greater than t = 0, which of the following is true: At time tB, both trains have the same velocity. Both trains speed up all the time. Both trains have the same velocity at some time before tB, . Somewhere on the graph, both trains have the same acceleration.

  13. Average Acceleration Change in instantaneous velocity divided by the time interval The x-component of the average acceleration

  14. Instantaneous Accelerationand Differentiation • For each time interval , calculate the x-component of the average acceleration • Take limit as sequence of the x-component average accelerations • The limiting value of this sequence is x-component of the instantaneous acceleration at the time t .

  15. Instantaneous Acceleration The x-component of acceleration is equal to the slope of the tangent line of the graph of the x-component of the velocity vs. time at time t

  16. Table Problem: Model Rocket A person launches a home-built model rocket straight up into the air at y = 0 from rest at time t = 0 . (The positive y-direction is upwards). The fuel burns out at t = t0. The position of the rocket is given by with a0 and g are positive. Find the y-components of the velocity and acceleration of the rocket as a function of time. Graph ayvs t for 0 < t < t0.

  17. Non-Constant Accelerationand Integration

  18. Velocity as the Integral of Acceleration The area under the graph of the x-component of the acceleration vs. time is the change in velocity

  19. Position as the Integral of Velocity Area under the graph of x-component of the velocity vs. time is the displacement Ei is the error in the approximation for each interval

  20. Summary: Time-Dependent Acceleration • Acceleration is a non-constant function of time • Change in velocity • Change in position

  21. Table Problem: Sports Car At t = 0 , a sports car starting at rest at x = 0accelerates with an x-component of acceleration given by and zero afterwards with • Find expressions for the velocity and position vectors of the sports as functions of time for t >0. (2) Sketch graphs of the x-component of the position, velocity and acceleration of the sports car as a function of time for t >0

  22. Concept Question A particle, starting at rest at t = 0, experiences a non-constant acceleration ax(t) . It’s change of position can be found by • Differentiating ax(t) twice. • Integrating ax(t) twice. • (1/2) ax(t) times t2. • None of the above. • Two of the above.

  23. Next Reading Assignment: Young and Freedman: Sections 3.1-3.3, 3.5

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