Collisions in Two Dimensions — Question 5

Question 5

A smooth contact transfers only the normal velocity component

Two identical smooth spheres collide elastically on a frictionless plane. At contact the unit vector from sphere 1 to sphere 2 is $\hat n=(3/5,4/5)$. Initial velocities are $\vec u_1=(5,0)$ and $\vec u_2=0\,\mathrm{m/s}$. Ignore spin.

1. Find both final velocity vectors by resolving along and perpendicular to $\hat n$.

2. Check momentum, kinetic energy and separation along the contact normal.

Question 5 - Solution

A smooth contact transfers only the normal velocity component

(a) Derivation and calculation

Project the incident velocity onto the contact normal: $$u_{1n}=\vec u_1\cdot\hat n=(5,0)\cdot(3/5,4/5)=3\ \mathrm{m/s}.$$ Equal masses exchange normal components while smooth contact leaves tangential components unchanged: $$\boxed{\vec v_2=3\hat n=(9/5,12/5)},\qquad\boxed{\vec v_1=(5,0)-3\hat n=(16/5,-12/5)\ \mathrm{m/s}}.$$

(b) Result and interpretation

Check all three requirements: $$\vec v_1+\vec v_2=(5,0)=\vec u_1,\qquad v_1^2+v_2^2=16+9=25=u_1^2.$$ $$(\vec v_2-\vec v_1)\cdot\hat n=(-7/5,24/5)\cdot(3/5,4/5)=3>0.$$ Thus momentum and energy are conserved, and the surfaces separate after contact.

Physical check

The contact impulse is parallel to the line joining the centers.