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Example: Volkwagen-Cadillac Scattering

Let's do a simple example in one dimension [thus avoiding the complications of adding and subtracting vectors] based on an apocryphal but possibly true story: A Texas Cadillac dealer once ran a TV ad showing a Cadillac running head-on into a parked Volkswagen Bug at 100 km/h. Needless to say, the Bug was squashed flat. Figs. 11.1 and 11.2 show a simplified sketch of this event, using the ``before-and-after'' technique with which our new paradigm works best. Figure 11.1 shows an elastic collision, in which the cars bounce off each other; Figure 11.2 shows a plastic collision in which they stick together.

  

Figure: Sketch of a perfectly elastic collision between a Cadillac initially moving at 100 km/h and a parked Volkswagen Bug. For an elastic collision, the magnitude of the relative velocity between the two cars is the same before and after the collision. [The fact that the cars look ``crunched'' in the sketch reflects the fact that no actual collision between cars could ever be perfectly elastic; however, we will use this limiting case for purposes of illustration.]

  

Figure: A perfectly inelastic or plastic collision in which the cars stick together and move as a unit after the collision.

For quantitative simplicity we assume that the Cadillac has exactly twice the mass of the Bug (M = 2m). In both cases the net initial momentum of the ``Caddy-Bug system'' is   M V<sub>i</sub> = 200 m,  where I have omitted the ``km/h'' units of V_i, the initial velocity of the Caddy. Therefore, since all the forces act between the components of the system, the total momentum of the system is conserved and the net momentum after the collision must also be 200 m.

In the elastic collision, the final relative velocity of the two cars must be the same as before the collision [this is one way of defining such a collision]. Thus if we assume (as on the drawing) that both cars move to the right after the collision, with velocities  V_f  for the Caddy and  v_f  for the Bug, then

$$v_f - V_f = 100 \qquad \hbox{\rm or} \qquad v_f = V_f + 100 .$$

Meanwhile the total momentum must be the same as initially:

$$M V_f \; + \; m v_f \; = \; 200 m \qquad \hbox{\rm or}$$

$$2 m V_f \; + \; m (V_f + 100) \; = \; 200 m$$

$$\hbox{\rm or} \qquad 3 m V_f = 100 m$$

giving the final velocities

$$V_f = 33 {1\over3} \; \hbox{\rm km/h} \qquad \hbox{\rm and} \qquad v_f = 133 {1\over3} \; \hbox{\rm km/h} .$$

In the plastic collision, the final system consists of both cars stuck together and moving to the right at a common velocity  v_f. Again the total momentum must be the same as initially:

$$(M + m) v_f = 200 m \qquad \hbox{\rm or}$$

$$3 m v_f = 200 m \qquad \hbox{\rm or}$$

$$v_f = 66 {2\over3} \; \hbox{\rm km/h} .$$

Several features are worth noting: first, the final velocity of the Bug after the elastic collision is actually faster than the Caddy was going when it hit! If the Bug then runs into a brick wall, well . . . . For anyone unfortunate enough to be inside one of the vehicles the severity of the consequences would be worst for the largest sudden change in the velocity of that vehicle -- i.e. for the largest instantaneous acceleration of the passenger. This quantity is far larger for both cars in the case of the elastic collision. This is why ``collapsibility'' is an important safety feature in modern automotive design. You want your car to be completely demolished in a severe collision, with only the passenger compartment left intact, in order to minimize the recoil velocity. This may be annoyingly expensive, but it is nice to be around to enjoy the luxury of being annoyed!

Back to our story: The Cadillac dealer was, of course, trying to convince prospective VW buyers that they would be a lot safer in a Cadillac - which is undeniable, except insofar as the Bug's greater maneuverability and smaller ``cross-section'' [the size of the ``target'' it presents to other vehicles] helps to avoid accidents. However, the local VW dealer took exception to the Cadillac dealer's stated editorial opinion that Bugs should not be allowed on the road. To illustrate his point, he ran a TV ad showing a Mack truck running into a parked Cadillac at 100 km/h. The Cadillac was quite satisfactorily squashed and the VW dealer suggested sarcastically that perhaps everyone should be required by law to drive Mack trucks to enhance road safety. His point was well taken.