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340 CHAPTER 11 Collisions
FIGURE 11.1 Crash test of a Mercedes–Benz automobile.
The photographs show an impact at 49 km h into a rigid barrier.
The first photograph was taken 5 10 3 s after the initial contact;
the others were taken at intervals of 20 10 3 s. The automobile
remains in contact with the barrier for 0.120 s; it then recoils from
the barrier with a speed of 4.7 km h. The checkered bar on the
ground has a length of 2 m.
According to this equation, the x component of the impulse for the force plotted in Fig.
11.2 is the area between the curve F (t) and the t axis.
x
The SI units of impulse are N s, or kg m s; these units are the same as those for
momentum.
By means of the equation of motion, F dp dt, we can transform Eq. (11.1) into
¢t ¢t dp
I F dt dt dp p p (11.2)
0 0 dt
where p is the momentum of the body before the collision (at time 0) and p is the
momentum after the collision (at time t t). Thus, the impulse of a force is simply
equal to the momentum change produced by this force. This equality of impulse and
momentum change is sometimes referred to as the impulse–momentum relation.However,
since the force acting during a collision is usually not known in detail, Eq. (11.2) is
not very helpful for calculating momentum changes. It is often best to apply Eq. (11.2)
in reverse, for calculating the time-average force from the known momentum change.
This time-average force is defined by
1 ¢t
F F dt (11.3)
¢t
0
In a plot of force vs. time, such as shown in Fig. 11.2, the time-average force simply
represents the mean height of the function above the t axis; this mean height is shown
F x
20 40 60 80 100 120 10 –3 s
0 t
N time-average
-
force e
–2 10 5
Force is zero
after 0.120 s.
–4 10 5
–6 10 5 Force varies as the
front end of the
automobile crumples.
FIGURE 11.2 Force on the automobile as a function of time during the impact
shown in Fig. 11.1. The colored horizontal line indicates the time-average force.
(Calculated from data supplied by Mercedes–Benz of North America, Inc.)
