I What is the exponential decay equation for a bouncing ball?

[wolf1728]

This is not a physics question.
Each time a ball bounces it will bounce to, let's say 75% of its previous height.
(I am not interested in the time, energy or velocity, of the ball.)
So if we drop it from 100 cm it will bounce back up to 75 cm, and on the next bounce it goes up to 56.25 cm and so on.
Here is a table for 6 bounces.
Starting height 100 cm
1st Bounce 75 cm
2nd Bounce 56.25 cm
3rd Bounce 42.1875 cm
4th Bounce 31.640625 cm
5th Bounce 23.73046875 cm
6th Bounce 17.797851563cm

Okay we can also say that on each bounce, the ball will lose 25% of its previous height.
So, basically, if we calculate this for every bounce, we can determine the height of any bounce.
Third Bounce Height = 100 -(100 *.25) - (75*.25) -(56.25*.25)

Finally, is there a much more compact formula then this one?
Thank you.

 

[kuruman]

h0= 100
h1 = 0.75 h0
h2 = 0.75 h1 = (0.75)² h0
h3 = 0.75 h2 = (0.75)³ h0
...
hn = ?

 

[wolf1728]

h0= 100
h1 = 0.75 h0
h2 = 0.75 h1 = (0.75)² h0
h3 = 0.75 h2 = (0.75)³ h0
...
hn = ?

Thanks for the quick reply!
Just to help you out, the coding for superscripts is ²
OR if you want to make the "squared" sign hold the "ALT" key and simultaneously type 0178 on the number key pad and you'll get ²
to get the "cubed" sign it's the "ALT" key plus 0179 and you get ³

 

[PeroK]

Thanks for the quick reply!
Just to help you out, the coding for superscripts is ²
OR if you want to make the "squared" sign hold the "ALT" key and simultaneously type 0178 on the number key pad and you'll get ²
to get the "cubed" sign it's the "ALT" key plus 0179 and you get ³

You can use Latex, which generally better:

https://www.physicsforums.com/help/latexhelp/

For example:

$h_n = (0.75^n) h_0$

 

[kuruman]

Thanks for the quick reply!
Just to help you out, the coding for superscripts is ²
OR if you want to make the "squared" sign hold the "ALT" key and simultaneously type 0178 on the number key pad and you'll get ²
to get the "cubed" sign it's the "ALT" key plus 0179 and you get ³

Thanks. I was too lazy to fuss with subscripts or LaTeX, nevertheless the message got across.

 

[Mark44]

(0.75)² h0

Just to help you out, the coding for superscripts is ²

Not at this site. The BBCode for exponents is ^exponent; i.e., with brackets, not angle brackets. This is what @kuruman used.

OR if you want to make the "squared" sign hold the "ALT" key and simultaneously type 0178 on the number key pad and you'll get ²
to get the "cubed" sign it's the "ALT" key plus 0179 and you get ³

Much too complicated and hard to remember, especially that you need to enter the numbers as octal, which is why you need to enter 0178 or 0179 instead of 178 or 179. As already advised, LaTeX works on this site, and is much nicer than BBCode or the extended ASCII codes for exponents of 2 and 3.

For tutorials for both BBCode and LaTeX, look under the INFO menu, under Help/How-to.

 

[wolf1728]

Thanks for the alert Mark44.
I've had my own website for about 20 years and I guess I haven't kept up with the latest "fancy-schmancy" computer tricks.
(Heck, my computer runs on coal! LOL)