Only Head On collision?

Let us consider two spheres A and B. Suppose they are interacting with each other (In broad sense one can say they are colliding). Let for the time being refer to "striking by coming in contact as collision". Now suppose the sphere B collided in such a way that its direction of motion is not along the line joining their centeres, in short it was not a head on collision. Then can this collision be an elastic one or is it so that there can be an elastic collision if and only if it is a head on collision? I hope the answer provider or commentator will justify his answer with reasons.

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Welcome to Physics.SE! The problem of glancing elastic collisions is treated exactly in most introductory textbooks, so yes they are possible. Could you clarify what you are confused about? – dmckee♦ Oct 13 '11 at 17:14
A little billiard-table physics should make the issues clear. – Mike Dunlavey Oct 13 '11 at 21:53

3 Answers

Head on collision is not required for elastic collision. Or the collision you described above can be an elastic collision.

To be an elastic collision, the momentum and kinetic energy should both be conserved, that is: assume the velocity for sphere 1 and 2 are $\vec{v_1}$ and $\vec{v_2}$ accordingly, and the direction of neither of them is along the direction of the line joining their centers, and assume the velocity after the collision are $\vec{v_1'}$ and $\vec{v_2'}$, then the elastic collision requires:

$m_1\vec{v_1} +m_2\vec{v_2} = m_1\vec{v_1'}+m_2\vec{v_2'}$

and

$\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2 v_2'^2$

apply some initial conditions: like the angle between two velocities etc, you can solve out the equation of motion of these two spheres after collision.

So, to sum up, if you want tell whether a collision is elastic or not, you just need to verify whether the motion of these spheres satisfy the two equations above. Hope this solves your problem.

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You haven't explicitly stated the nature of the contact between the spheres (i.e. is it frictionless?)

If we have ideal frictionless contact between the spheres, then yes the collision can be elastic regardless of whether it is head-on or glancing.

In the real world, with friction, then it does matter whether it is head on or not. For glancing contact, there will be slippage of the contact point during the collision, resulting in some energy dissipation. Also, this glancing contact will torque the spheres, transferring translational kinetic energy into rotational energy about their own centers of mass.

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you had written inelastic where the context meant elastic in the second paragraph and I edited it. – anna v Oct 14 '11 at 5:25

Elastic collision only implies that the collision conserves momentum as well as energies(in the form of kinetic energy) i.e. $$m_1u_1+m_2u_2 = m_1v_1+m_2v_2$$ and $$\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$$

Where as in inelastic collision, only momentum is conserved while the energy( as in kinetic energy) is not conserved, which is lost as heat, sound, light etc.

It doesn't matter if the collision is head on or oblique for the collision to be classified as elastic or inelastic.

regards,

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