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Using the Derivative

Using the Derivative. AP Physics C Mrs. Coyle. http://www.ima.umn.edu/~arnold/graphics.html. Instantaneous Velocity. v = lim D x D t 0 D t. or. v = d x dt. Instantaneous Acceleration. a = lim D v D t 0 D t. or. a = dv dt.

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Using the Derivative

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  1. Using the Derivative AP Physics C Mrs. Coyle http://www.ima.umn.edu/~arnold/graphics.html

  2. Instantaneous Velocity v = limDx Dt0Dt or v = dx dt

  3. Instantaneous Acceleration a = limDv Dt0Dt or a = dv dt

  4. Using the limit to calculate instantaneous acceleration. • Example 1: The velocity of a particle is given by v= -t2 + 2 (t is in sec). Find the instantaneous acceleration at t= 4s (using the limit). Answer: -8 m/s

  5. Evaluating the derivative of a polynomial. For y(x) = axn dy = a n xn-1 dx -Apply to each term of the polynomial. -Note that the derivative of constant is 0.

  6. Using the derivative to calculate instantaneous acceleration. • Example 2: The velocity of a particle is given by v= -t2 + 2 (t is in sec). Find the instantaneous acceleration at t= 4s (using the derivative). Answer: -8 m/s

  7. Example 3: A particle’s position is given by the expression x= 4-t2 + 2t3 (t is in sec). Find for t= 5s : • Its position • Its velocity • Its acceleration Answer: a) 229m, b) 140 m/s, c) 58 m/s2

  8. Example 4 An object follows the equation of motion x= 3t2 -10t +5. • At what time(s) is its position equal to zero? • At what time is its velocity equal to zero? Hint: Remember for a quadratic equation ax² + bx + c = 0 , the roots are: Answer: a) 0.62sec and 2.7sec, b) 1.7sec

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