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Modelling of ball bounce

Kaspelek

New member
Apr 21, 2013
26
Hi guys, just an intuitive question i've come across. Quite ambiguous, not sure on the correct response.

So basically i'm given a scenario where I'm provided the data of an actual height vs time points of a vertical ball drop and it's bounce up and back down etc.

Question starts off where I have to model the height and time of the bounce using the points given to the equation f(x)=|a*sin(b(x-c))|

Hence i work out a, b and c a=0.5, b=3, c=0.6 and drew the graph.

Commented on the fit of the model.


Next I am asked to create a new function s(x) where it is created by multiplying the original f(x) function by an exponential function e(x) i.e. s(x)=e(x)*f(x).

This provides a decaying effect of the height, hence more realistic.

Finally, I am asked to draw a new function, h(x), whereby i only remove the value a from the f(x) function so h(x)=e(x)*|sin(b(x-c))|.

I am then asked, Having removed a=0.5 in f(x), why does this give a more accurate model than s(x).


My thoughts?
I believe it is because that since the value of a is less than 1, the height of the ball bounce is proportionally decreasing unnecessarily when comparing the h(x) and s(x) models respectively.




Thoughts guys?


Thanks in advance.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
By the time you get to the first bounce, the decaying amplitude (which should presumably be $e^{-x}$) is:

\(\displaystyle e^{-0.6}\approx0.55\)

and this is closer to 0.5 than half that value.
 

Kaspelek

New member
Apr 21, 2013
26
Suggested that the response is worth 2 marks, thinking there's more.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
What has the amplitude decayed to when the ball reaches it's peak after the first bounce? Unless you know some differential calculus, you will have to rely on a graph...