How Does Removing the Amplitude Factor Improve Ball Bounce Modelling?

[Kaspelek]

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]

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]

Suggested that the response is worth 2 marks, thinking there's more.

 

[MarkFL]

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...

 

[CellCoree]

I would first commend the person for their intuitive approach to the problem and their use of mathematical modeling to understand the behavior of the ball bounce. I would also agree with their reasoning for why removing the value of a from the function h(x) gives a more accurate model.

However, I would also suggest that the person should consider the physical factors that may affect the ball bounce, such as air resistance, surface friction, and the elasticity of the ball. These factors can also be incorporated into the model to make it more realistic and accurate. Additionally, it would be helpful to compare the modeled results with experimental data to further validate the accuracy of the model.

Overall, I would encourage the person to continue exploring and refining their model to better understand the behavior of the ball bounce. This type of problem-solving approach is essential in the field of science and can lead to new discoveries and advancements.