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 $F_a$.



  • $\begingroup$ What is a "bounce factor"? Is it a percentage of energy lost in the bounce? $\endgroup$ – Jerry Schirmer Mar 28 '12 at 21:38
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    $\begingroup$ @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. $\endgroup$ – Garmen1778 Mar 28 '12 at 21:40
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    $\begingroup$ 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? $\endgroup$ – David Z Mar 28 '12 at 22:03
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    $\begingroup$ @DavidZaslavsky: The coefficient of restitution is the answer--- why not post it as such? $\endgroup$ – Ron Maimon Mar 29 '12 at 3:04
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    $\begingroup$ @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. $\endgroup$ – Manishearth Mar 29 '12 at 4:50

I've also answered a similar question here.


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 $y$.


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.


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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