Figure 8.11A two-dimensional collision with the coordinate system chosen so thatm 2 is initially at rest andv 1 is parallel to thex-axis. This coordinate system is
sometimes called the laboratory coordinate system, because many scattering experiments have a target that is stationary in the laboratory, while particles are scattered from it
to determine the particles that make-up the target and how they are bound together. The particles may not be observed directly, but their initial and final velocities are.
Along thex-axis, the equation for conservation of momentum is
p 1 x+p 2 x=p′ (8.58)
1 x+p′ 2 x.
Where the subscripts denote the particles and axes and the primes denote the situation after the collision. In terms of masses and velocities, this
equation is
m 1 v 1 x+m 2 v 2 x=m 1 v′ 1 x+m 2 v′ 2 x. (8.59)
But because particle 2 is initially at rest, this equation becomes
m 1 v 1 x=m 1 v′ 1 x+m 2 v′ 2 x. (8.60)
The components of the velocities along thex-axis have the formvcosθ. Because particle 1 initially moves along thex-axis, we findv 1 x=v 1.
Conservation of momentum along thex-axis gives the following equation:
m 1 v 1 =m 1 v′ 1 cosθ 1 +m 2 v′ 2 cosθ 2 , (8.61)
whereθ 1 andθ 2 are as shown inFigure 8.11.
Conservation of Momentum along thex-axis
m 1 v 1 =m 1 v′ 1 cosθ 1 +m 2 v′ 2 cosθ 2 (8.62)
Along they-axis, the equation for conservation of momentum is
p 1 y+p 2 y=p′ 1 y+p′ 2 y (8.63)
or
m 1 v 1 y+m 2 v 2 y=m 1 v′ 1 y+m 2 v′ 2 y. (8.64)
Butv 1 yis zero, because particle 1 initially moves along thex-axis. Because particle 2 is initially at rest,v 2 yis also zero. The equation for
conservation of momentum along they-axis becomes
0 =m 1 v′ 1 y+m 2 v′ 2 y. (8.65)
The components of the velocities along they-axis have the formvsinθ.
Thus, conservation of momentum along they-axis gives the following equation:
0 =m 1 v′ 1 sinθ 1 +m 2 v′ 2 sinθ 2. (8.66)
Conservation of Momentum along they-axis
0 =m 1 v′ 1 sinθ 1 +m 2 v′ 2 sinθ 2 (8.67)
CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS 277