5.3: Head-on Collision of a Moving Sphere with an Initially Stationary Sphere
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The coefficient of restitution is
e=relative speed of recession after collisionrelative speed of approach before collision.
We suppose that the two masses m1 and m2, the initial speed u, and the coefficient of restitution e are known; we wish to find v1 and v2.
We evidently need two equations. Since there are no external forces on the system, the linear momentum of the system is conserved:
m1u=m1v1+m2v2.
The second equation will be the restitution equation (Equation 5.3.1):
v2−v1=eu.
These two equations can be solved to yield
v1=(m1−m2em1+m2)u
and
v2=(m1(1+e)m1+m2)u.The relation between the kinetic energy loss and the coefficient of restitution isn't quite as simple as in Section 5.2.
Show that
kinetic energy (after)kinetic energy (before)=m1v21+m1v22m1u2=m1+m2e2m1+m2.
- If m2=∞ (as in Section 5.2), this becomes just e2. If e=1, it becomes unity, so all is well.
- If m1<<m2 (Ping-pong ball collides with cannon ball), v1=−u,v2=0.
- If m1=m2 (Ping-pong ball collides with ping-pong ball), v1=0,v2=u.
- If m1>>m2 (Cannon ball collides with ping-pong ball), v1=u,v2=2u.
A moving sphere has a head-on elastic collision with an initially stationary sphere. After collision the kinetic energies of the two spheres are equal. Show that the mass ratio of the two spheres is 0.1716.
Which of the two spheres is the more massive? (I guarantee that your answer to this will be correct.)