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11.4 Collisions in Two and Three Dimensions 351

11.4 COLLISIONS IN TWO

AND THREE DIMENSIONS

In the previous sections, we have focused on collisions on a straight line, in one dimen-
sion. Collisions in two or three dimensions are more difficult to analyze, because the
conservation laws for momentum and energy do not provide sufficient information to
determine the final velocities completely in terms of the initial velocities. Momentum
is always conserved during a collision, and this conservation provides one equation for
each of the x, y, and z directions. If it is known that the collision is totally elastic, then
conservation of the total kinetic energy provides another equation. However, these are
not enough to determine the three final velocity components for each and every particle.
Some information concerning the final velocities must also be known or measured.
The case of totally inelastic collisions is an exception: in this case, the conserva-
tion of momentum determines the outcome completely, even in two or three dimen-
sions.The particles stick together, and their final velocities coincide with the velocity
of the center of mass, as illustrated by the following example. The subsequent exam-
ple explores a case where the solution exploits some knowledge of the final velocities.

A red automobile of mass 1100 kg and a green automobile of
EXAMPLE 7 Concepts
mass 1300 kg collide at an intersection. Just before this collision, in
Context
the red automobile was traveling due east at 34 m s, and the green automobile was
traveling due north at 15 m s (see Fig. 11.9). After the collision, the wrecked auto-
mobiles remain joined together, and they skid on the pavement with locked wheels.
What is the direction of the skid?
SOLUTION: The final velocity of the wreck coincides with the final velocity of
the center of mass, which is the same as the initial velocity of the center of mass.
According to Eq. (10.37), this velocity is
m v m v final velocity in totally
2 2
1 1
v (11.27)
CM inelastic collision
m m
1 2
With the x axis eastward and the y axis northward, the initial velocity v of the red
1
automobile has an x component but no y component, and the initial velocity v of
2
the green automobile has a y component but no x component. Hence the x com- Wrecked vehicles lock
together and move off
ponent of v is
CM at this angle q.
m v 1100 kg 34 m s y
1 1
v 16 m s v
CM,x CM
m m 1100 kg 1300 kg
1 2
and the y component of v is
CM q
m v 1300 kg 15 m s v 1 O
2 2
v 8.1 m s x
CM, y
m m 1100 kg 1300 kg
1 2
Before collision, v 2
The angle between the direction of this velocity and the x axis is vehicle velocities
given by are perpendicular.
v CM, y 8.1 m s
tanu 0.51
v 16 m s
CM, x
from which
FIGURE 11.9 An automobile collision.
u 27 Before the collision, the velocities of the
automobiles were v and v . After the colli-
2
1
Since the x axis is eastward, this is 27 north of east. sion, both velocities are v .
CM