# What are all the equations we use to calculate how bounces work?

I mean, what is the object's final displacement, or the function that describes the object's height over time (see [1]) of an object thrown by a height $h$ with a speed of $\vec{v_0}$, a mass of $m$, a "bounce factor" of $\Lambda$ and the floor with a friction of $F_f$ and the air with a friction of $F_a$.

[1]

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What is a "bounce factor"? Is it a percentage of energy lost in the bounce? – Jerry Schirmer Mar 28 '12 at 21:38
@JerrySchirmer I put it into "" because I don't know what is it, but there must be something like that. A percentage of the energy lost in the bounce maybe. – Garmen1778 Mar 28 '12 at 21:40
It's called the coefficient of restitution, but anyway... I don't think this question is well formulated because it's not about a physical concept, you're just asking for a list of equations (which, by the way, can be found in any intro mechanics textbook as well as on thousands of websites, including probably about a dozen Wikipedia pages alone). I'm not sure it really warrants closing, so I won't (for now), but could you try explaining what research you've done and/or what you've tried and what you're still looking for afterwards? – David Z Mar 28 '12 at 22:03
@DavidZaslavsky: The coefficient of restitution is the answer--- why not post it as such? – Ron Maimon Mar 29 '12 at 3:04
@DavidZaslavsky: While I do agree that this is sort-of-borderline-too-localised, I don't think that this is easily solved via The Google. I've never really seen lists of usable formulae anywhere, and if the OP needs this for, say a game, then finding all the relevant formulae will be tough. – Manishearth Mar 29 '12 at 4:50

I've also answered a similar question here.

## Variables

I'm using the subscript $y$ to denote stuff in the perpendicular direction (along the $y$ or $h$ axis), and $x$ for stuff in the parallel direction (along $x$).

I'll use $u$ for initial velocities and $v$ for final velocities. The initial and final refer to "just before/after a bounce", and "just before/after an arc", where "bounce" refers to the moment when it touches the ground, and "arc" is the arcing motion afterwards.

$e$ is the coefficient of restitution--this is the "bounciness" you wanted. It can have any value from 0 to 1, where 0 is completely unbouncy(inelastic), and 1 is very bouncy. (elastic). A value greater than one gives an unphysical effect where it bounces higher. It's related to energy via $\text{loss in energy} = \frac12mv^2(1-e^2)$, for a ball bouncing on the ground. So I guess $\Lambda_\text{(percentage of energy loss)}=1-e^2\times100\%$

$\mu$ is the friction coefficient for the ground-ball interface. It is 0 for a frictionless surface, and usually less than 1. Not necessarily, though. It will lead to an additional loss of energy not taken into account by $e$ or $\Lambda$.

$g$ is gravitational acceleration

Note that all the quantities are signed.

## Relevant formulae, condensed version

### For a bounce

$$v_y=-eu_y$$ $$v_x=u_x+ \mu(e-1)u_y$$

These $v$s become $u$s for the upcoming arc.

Bounces are pretty much instantaneous. If you want to consider the time factor, you need to know the shape and young's modulus of the object.

### For an arc

The arc will be executed in a time $t_{arc}=2u_y/g$, and will attain a maximum height $y_{max}=\frac{u^2}{2g}$

During this time: ($t$ is the time since the arc started, NOT the total time) $$y=u_y-\frac12gt^2$$ $$x=u_xt$$

At the moment the arc finishes,

$$v_x=u_x$$ $$v_y=-u_y$$ These $v$s become the $u$s for the next "bounce"

## All the forumulae

### SUVAT equations

See . In this case, $a_y=-g,a_x=0$, and $s$ is distance travelled in relevant direction. You can apply these equation separately for $x$ and $y$.

### Bouncing

Here, $N$ is normal force. $f$ is the friction force. $J$ refers to impulse, $p$ to momentum. Friction $$f=\pm\mu N \text{ direction can vary}$$ Impulse $$J_y=\int N\rm dt$$ $$J_x=\int f\rm dt$$ We can combine these to get $J_x=\pm\mu J_y$ We take the sign $-$ in this case, as friction opposes motion, and the motion in $x$ direction is positive.

$$J=\Delta p=m(v-u) \text{ for both axes}$$

Combining all these, you can get the bounciness equations.

Note: if you want your ball to have a spin as well, the equations become more complicated.

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I would use a combination of projectile motion and restitution/collision equations for a simple model. Model each bounce individually using projectile motion eqs. for trajectory and for each bounce collision, use restitution/collision equations to calculate the angle of launch for the next bounce as well as initial energy/velocity.

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