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Chapter 8 | Linear Momentum and Collisions 341
The components of the velocities along the -axis have the form . Because particle 1 initially moves along the -axis, we find .
Conservation of momentum along the -axis gives the following equation:
where and are as shown in Figure 8.14.
Along the -axis, the equation for conservation of momentum is
(8.79)
(8.81) (8.82)
Conservation of Momentum along the -axis
(8.80)
or
But is zero, because particle 1 initially moves along the -axis. Because particle 2 is initially at rest, is also zero. The
equation for conservation of momentum along the -axis becomes
The components of the velocities along the -axis have the form . Thus, conservation of momentum along the -axis gives the following equation:
The equations of conservation of momentum along the -axis and -axis are very useful in analyzing two-dimensional
(8.83)
(8.84)
Conservation of Momentum along the -axis
(8.85)
collisions of particles, where one is originally stationary (a common laboratory situation). But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level.
Making Connections: Real World Connections
We have seen, in one-dimensional collisions when momentum is conserved, that the center-of-mass velocity of the system remains unchanged as a result of the collision. If you calculate the momentum and center-of-mass velocity before the collision, you will get the same answer as if you calculate both quantities after the collision. This logic also works for two- dimensional collisions.
For example, consider two cars of equal mass. Car A is driving east (+x-direction) with a speed of 40 m/s. Car B is driving north (+y-direction) with a speed of 80 m/s. What is the velocity of the center-of-mass of this system before and after an inelastic collision, in which the cars move together as one mass after the collision?
Since both cars have equal mass, the center-of-mass velocity components are just the average of the components of the individual velocities before the collision. The x-component of the center of mass velocity is 20 m/s, and the y-component is 40 m/s.
Using momentum conservation for the collision in both the x-component and y-component yields similar answers:
(8.86)
(8.87) (8.88) (8.89)
Since the two masses move together after the collision, the velocity of this combined object is equal to the center-of-mass velocity. Thus, the center-of-mass velocity before and after the collision is identical, even in two-dimensional collisions, when momentum is conserved.
