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11.1 Impulsive Forces 341

by the red horizontal line in Fig. 11.2. According to Eq. (11.2), we can write the time-
average force as
1 1
F I (p p) (11.4) average force in collision
¢t ¢t

This relation gives a quick estimate of the average magnitude of the impulsive force
acting on the body if the duration of the collision and the momentum change are known.

The collision between the automobile and the barrier shown in
EXAMPLE 1 Concepts
Fig. 11.1 lasts 0.120 s.The mass of the automobile is 1700 kg, in
Context
and the initial and final velocities in the horizontal direction are v 13.6 m s and
x
v 1.3 m s, respectively (the final velocity is negative because the automobile
x
recoils, or bounces back from the barrier). From these data, evaluate the average force
that acts on the automobile during the collision. Evaluate the average force that
acts on the barrier.

SOLUTION: With the x axis along the direction of the initial motion, the change
of momentum is

p p mv mv
x x x x
1700 kg ( 1.3 m s) 1700 kg 13.6 m s

4
2.53 10 kg m s
According to Eq. (11.4), the average force is then

p p
x
F x
x
¢t
(11.5)
4
2.53 10 kg m s
5
2.11 10 N
0.120 s
Since the mutual forces on two bodies engaged in a collision are an action–reaction
pair, the forces on the automobile and on the barrier are of equal magnitudes and of
5
opposite directions. Thus, the average force on the barrier is F 2.11 10 N.
x
4
This is quite a large force—it equals the weight of about 2 10 kg, or 20 tons.
When an automobile collides with an obstacle and suddenly
EXAMPLE 2 Concepts
stops, a passenger not restrained by a seat belt will not stop in
Context
simultaneously with the automobile, but instead will continue traveling at nearly
constant speed until he or she hits the dashboard and the windshield.The collision
of the passenger’s head with the windshield often results in severe or fatal injuries.
In crash tests, dummies with masses, shapes, and joints simulating human bodies
are used to determine likely injuries. Consider a dummy head striking a wind-
shield at 15 m s (54 km h) and stopping in a time of 0.015 s (this time is consid-
erably shorter than the time of about 0.12 s for stopping the automobile because
the front end of the automobile crumples gradually and cushions the collision to
some extent; there is no such cushioning for the head striking the windshield).
What is the average force on the head during impact on the windshield? What is
the average deceleration? Treat the head as a body of mass 5.0 kg, moving inde-
pendently of the neck and trunk.