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310 CHAPTER 10 Systems of Particles

This is a direct consequence of the equality of the magnitudes of the action and reac-
tion forces that act on the shot and the gun during the firing. The force gives the
shot (of small mass) a large acceleration, and the reaction force gives the gun
(of large mass) a small acceleration.
In this calculation we neglected the mass and momentum of the gases released
in the explosion of the gunpowder.This extra momentum increases the recoil veloc-
ity somewhat.

An automobile of mass 1500 kg traveling at 24 m/s crashes
EXAMPLE 3
into a similar parked automobile.The two automobiles remain
joined together after the collision. What is the velocity of the wreck immediately
after the collision? Neglect friction against the road, since this force is insignifi-
cant compared with the large mutual forces that the automobiles exert on each
other.

SOLUTION: Under the assumptions of the problem, the only horizontal forces
are the mutual forces of one automobile on the other.Thus, momentum conservation
applies to the horizontal component of the momentum: the value of this compo-
nent must be the same before and after the collision. Before the collision, the
(horizontal) velocity of the moving automobile is v 24 m/s and that of the other
1
is v 0. With the x axis along the direction of motion (see Fig. 10.4), the total
2
momentum is therefore
P m v m v m v
x 1 1 2 2 1 1
After the collision, both automobiles have the same velocity (see Fig. 10.4b). We
will designate the velocities of the automobiles after the collision by v and v ,
2
1
respectively. We can write v v v (the automobiles have a common v , since
1
2
they remain joined), so the total momentum is
P m v m v (m m )v
1 1
2 2
2
1
x
PROBLEM-SOLVING TECHNIQUES CONSERVATION OF MOMENTUM

Note that the solution of these examples involves three steps momentum are conserved separately. Thus, before writing the
similar to those we used in examples of energy conservation: expressions for the momentum, you need to select coordinate
axes and decide which components of the momentum you
1 First write an expression for the total momentum P before
want to examine. If the motion is one-dimensional, place one
the firing of the gun or the collision of the automobiles.
axis along the direction of motion, such as the x axis in the
2 Then write an expression for the total momentum P
above examples. It then suffices to examine the x component
after the firing or the collision.
of the momentum.However,sometimes it is necessary to exam-
3 And then use momentum conservation to equate these ine two components of the momentum (or, rarely, three); then
expressions. two (or three) equations result.When writing the components
of the momentum, pay attention to the signs; the component
However,in contrast to energy conservation,you must keep
is positive if the motion is along the direction of the axis, neg-
in mind that momentum conservation applies to the compo-
ative if the motion is opposite to the direction of the axis.
nents of the momentum—the x, y, and z components of the