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How do you calculate the new velocity of the spheres after a collision when the spheres have different radii?

I thought you could just decompose the horizontal velocities and use the standard collision equation on the velocities on the axis of the collision like so.

Sphere vectors decomposed:

Decomposed Collision

Then just add $a'_1$ and $a_2$ and similarly $b'_1$ and $b_2$ to get the new velocity vectors.

To take it further if there is a condition that means the spheres cannot 'jump' off the surface i.e. its fixed to the surface how would that change the vector velocities.

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  • $\begingroup$ You imply they are purely sliding? When they are rolling the angular momentum might change the result. $\endgroup$
    – Juraj
    Commented Apr 19 at 21:33

1 Answer 1

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So here not only there are impulses between the spheres, but also between each sphere and the ground. The result is a system of equations with three expressions and three unknown impulses.

For the contact acting on the i-th body from the j-th body (or ground), consider the contact normal $\hat{n}_{ij}$, the impulse $J_{ij}$ and the contact conditions

$$ \hat{n}_{ij} \cdot ( \vec{v}_i^\star - \vec{v}_j^\star ) = -\epsilon_{ij}\, \hat{n}_{ij} \cdot ( \vec{v}_i - \vec{v}_j ) \tag{1}$$

where $\vec{v}_i$ is the velocity vector of the i-th body before impact and $\vec{v}_i^\star$ after impact. The same for $\vec{v}_j$. Finally, $\epsilon_{ij}$ is the coefficient of restitution for the contact.

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For the contact normal vectors above, I suppose the contacts with the ground have zero COR and the contact between the spheres has 1.

The result of all impacts to each body is

$$ \begin{aligned} \vec{v}_1^\star &= \vec{v}_1 + \frac{1}{m_1} \left( \hat{n}_{12} J_{12} + \hat{n}_{10} J_{10} \right) \\ \vec{v}_2^\star &= \vec{v}_2 + \frac{1}{m_2} \left( -\hat{n}_{12} J_{12} + \hat{n}_{20} J_{20} \right)\\ \end{aligned} \tag{2}$$

These two expressions are used in the following 3 contacts to make the system in terms of the three terms $J_{ij}$ only

$$\begin{aligned} \hat{n}_{12} \cdot ( \vec{v}_1^\star - \vec{v}_2^\star ) & = -\epsilon_{12}\, \hat{n}_{12} \cdot ( \vec{v}_1 - \vec{v}_2 ) \\ \hat{n}_{10} \cdot ( \vec{v}_1^\star ) & = -\epsilon_{10}\, \hat{n}_{10} \cdot ( \vec{v}_1 ) \\ \hat{n}_{20} \cdot ( \vec{v}_2^\star ) & = -\epsilon_{20}\, \hat{n}_{20} \cdot ( \vec{v}_2 ) \end{aligned}$$

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