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Writing a trigonometric function for a bouncing ball.

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Write a trigonometric function for a ball dropped from a distance of 5ft from the floor. Let the x-axis represent the time after the ball was dropped and the y axis represent the height in feet. Address these issues:
1. Explain the process you used to find the function. Include all math steps. Why did you select this particular equation?
2. What did you have to do to modify this equation to get it to fit to the behavior of the bouncing ball?
3. Graph the equation and label axes.

Assume in this model that the ball rebounds 3/4 of its height each bounce.

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Solution Summary

The trigonometric function for a bouncing ball is examined. All equations are shown.

Solution Preview

Let initial height is H ( 5 ft.) and ball is dropped: event A
before 1st bounce, height from the ground can be given as h,

h = H - (1/2) g t^2

so, when boll reaches the ground first time, say event B,

vb = sqrt(2gH)
and,
tb = sqrt(2H/g)

After first bounce,
let velocity of reflection of ball is decreased to e times before the collision,i.e,

ub = e.sqrt(2gH)
now,

h = e.sqrt(2gH) (t - sqrt(2H/g)) - 1/2 g (t - sqrt(2H/g))^2
and,
v = e.sqrt(2gH) - g (t - sqrt(2H/g))
Now, when it reaches to top, say event C
vc = ...

Solution provided by:
Amrit Lal Ahuja, PhD (IP)
Education
  • BEng, Allahabad University, India
  • MSc , Pune University, India
  • PhD (IP), Pune University, India
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