motion derivatives continued derivatives of trig functions sin’x =>cos xcos’x => - sin x...

MOTION

Derivatives Continued

Derivatives of Trig Functions• sin’x => cos xcos’x => - sin x• tan’x => sec2x cot’x => - csc2x• sec’x => sec x tan x csc’x => - csc x cot x

Remember what you already know!

Example of Harmonic Motion

• (jumping up and down)• (bouncing a ball)

• Example of weight on a spring.

• motion s = 5 cos t• velocity s’ = -5 sin t• acceleration s’’ = - 5 cos t

Time for an experiment.• You need: a motion

detector, TI-83 Plus, bouncing ball program, and a ball.

• Bounce the ball and collect the data using the program.

• Graph the “Time vs. Distance” graph.

• Run the regression equation for a sin graph.

Answer these questions:

1)In 5 seconds how many times did the ball hit the ground?

Height

time

2) What is the height of the ball at t=2?

3)At what time is the ball at it’s highest point?

(maximum)

4)Take the heights at each bounce and plot them in the stat plot, find the exponential equation for the heights.

5)Multiply the exponential equation by the sinusoidal equation that you got earlier and you will have the dampened equation that represents your ball exactly.

6)Using your new equation, find the first derivative and the second derivative.

7)What does the second derivative represent?

8)At what time was the balls velocity the greatest?

9)At what time was the balls speed the greatest?

10)At what time or times was the balls velocity equal to zero? What does this mean and what is happening to the ball at these times?

11)What is the limit as t approaches infinity?