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PH 201

PH 201. Dr. Cecilia Vogel Lecture 3. REVIEW. Motion in 1-D instantaneous velocity and speed acceleration. OUTLINE. Graphs Constant acceleration x vs t, v vs t, v vs x Vectors notation magnitude and direction. Sign of Acceleration. Mathematically

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PH 201

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  1. PH 201 Dr. Cecilia Vogel Lecture 3

  2. REVIEW • Motion in 1-D • instantaneous velocity and speed • acceleration OUTLINE • Graphs • Constant acceleration • x vs t, v vs t, v vs x • Vectors • notation • magnitude and direction

  3. Sign of Acceleration • Mathematically • If (signed) velocity increases, a is + • If (signed) velocity decreases, a is - • Memorize • If velocity and acceleration are in same direction, object will speed up • If velocity and acceleration are in opposite directions, object will slow down • Physical intuition • positive acceleration produced by push or pull in + direction • negative acceleration produced by push or pull in - direction

  4. Position, Velocity, Acceleration • Velocity is • slope of tangent line on an x vs t graph • limit of Dx/Dt as Dt goes to zero • the derivative of x with respect to time • dx/dt • Similarly acceleration is • slope of tangent line on a v vs t graph • limit of Dv/Dt as Dt goes to zero • the derivative of v with respect to time • dv/dt • If you have position as a function of time, x(t) • can take derivative to find v(t) • take derivative again to find a(t)

  5. Derivatives of Polynomials • The derivative with respect to time of a power of t, if C is a constant: • Special case, if the power is zero: • The derivative of a sum is sum of derivatives: • ex

  6. Example • ex • The acceleration at t=0 is -6 m/s2, and at t=3 is 90 m/s2. • The average acceleration between t=0 and t=3 is 39 m/s2

  7. Special Case: Constant Velocity • Acceleration is zero • Graph of x vs. t is linear • slope is constant • Average velocity is equal to the constant velocity value, v becomes if initial time is zero, and we drop subscript on final variables.

  8. Special Case: Constant Acceleration • If object’s acceleration has a constant value, a, • then its velocity changes at a constant rate: • And its position changes quadratically with time:

  9. Position with Constant Acceleration • Slope of the position graph (velocity) is constantly changing • quadratic function of time.

  10. Example A little red wagon is rolling in the positive direction with an initial speed of 5.0 m/s. A child grabs the handle and pulls, giving it a constant acceleration of 1.1 m/s2 opposite its initial motion. Let the time the child begins to pull be t=0, and take the position of the wagon at that time to be x=0. How fast will the wagon be going after 1.0 s of pulling? Where will the wagon be then? At what time will the wagon come to a stop (for an instant)?

  11. What if…? • What if I asked “where will the wagon be when it is going -1.0 m/s?” • You could: • find the time that v= -1.0 m/s • find the position at that time.

  12. What if…? • Let’s find a generalization of that: • Where will object be when it’s velocity is v, given a known initial position, velocity, and constant acceleration? simplify:

  13. Derivatives and Constant Acceleration Yeah – consistency!

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