8.6 Collisions of point masses in two dimensions  (Page 4/6)

Because the masses are equal, $m_1=m_2=m$ . Algebraic manipulation (left to the reader) of conservation of momentum in the $x$ - and $y$ -directions can show that

(Remember that ${\theta }_2$ is negative here.) The two preceding equations can both be true only if

There are three ways that this term can be zero. They are

• ${v\prime }_1^{}=0$ : head-on collision; incoming ball stops
• ${v\prime }_2^{}=0$ : no collision; incoming ball continues unaffected
• $\text{cos}({\theta }_1-{\theta }_2)=0$ : angle of separation $({\theta }_1-{\theta }_2)$ is $\text{90º}$ after the collision

All three of these ways are familiar occurrences in billiards and pool, although most of us try to avoid the second. If you play enough pool, you will notice that the angle between the balls is very close to $\mathrm{90º}$ after the collision, although it will vary from this value if a great deal of spin is placed on the ball. (Large spin carries in extra energy and a quantity called angular momentum , which must also be conserved.) The assumption that the scattering of billiard balls is elastic is reasonable based on the correctness of the three results it produces. This assumption also implies that, to a good approximation, momentum is conserved for the two-ball system in billiards and pool. The problems below explore these and other characteristics of two-dimensional collisions.

Test prep for ap courses

Section summary

• The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the $x$ -axis parallel to the velocity of the incoming particle.
• Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the $x$ -axis), stated by $m_1{v}_1=m_1{v\prime }_1\text{cos}{\theta }_1+m_2{v\prime }_2\text{cos}{\theta }_2$ and along the direction perpendicular to the initial direction (the $y$ -axis) stated by $0=m_1{v\prime }_{1y}+m_2{v\prime }_{2y}$ .
• The internal kinetic before and after the collision of two objects that have equal masses is
  $\frac{1}{2}{{\text{mv}}_1}^2=\frac{1}{2}{{\text{mv}\prime }_1}^2+\frac{1}{2}{{\text{mv}\prime }_2}^2+{\text{mv}\prime }_1{v\prime }_2\text{cos}\left({\theta }_1-{\theta }_2\right).$
• Point masses are structureless particles that cannot spin.