O o

Figure 7.5. The free fall of a mass (mA) onto a mass-spring system with mass mB.

v where mA is the falling mass, mB is the mass attached to the spring, vo is the velocity of the center of mass of object A just before impact, and Vf is the velocity of the combination after the impact. Because the mass A falls from a distance h its velocity before the collision is given by the equation:

in which g denotes as usual the magnitude of the gravitational acceleration and e2 is the unit vector in the direction of gravitational force. The velocity of the two masses A and B right after the impact is then given by the following relation:

Example 7.2. Quarterback Hits a Concrete Wall. Gus Ferrotte, the former quarterback of the Washington Redskins, scored a touchdown against the New York Giants in 1997 and then in the excitement of the moment roamed head-on to the nearby concrete wall (Fig. 7.6). Gus had a helmet on but nevertheless strained his neck. He also became a subject of jokes. To explain his behavior, he said to a television reporter, "I would hit heads with other football players after a touchdown and it never caused strain in my neck." Is there a difference between hitting a football player on the head and hitting a concrete wall?

Solution: Let us assume that Gus is represented by an object of mass m1, and the football player he would hit head on by m2. The masses m1 and m2 are comparable in magnitude. Assuming that Gus had a velocity vo e1 before the collision and the other player was at rest and further assuming that after Gus hits him, they move in unison with velocity v e1, we find that:

Figure 7.6. A quarterback running toward a concrete wall with the intention of hitting it head-on in celebration of his touchdown.

If the two football players were about the same weight, Gus's speed would decrease by one-half as a result of the collision. Thus, according to Eqn. 7.3, the impulse acting on him is given by the following relation:

Next, let us represent the concrete wall as an object with mass M. Because M is much greater than mi, when Gus hits the concrete wall, he loses all his linear momentum during collision; hence, in this case

Thus, the impulse of collision with a concrete wall is twice as great as that of colliding head-on with another player.

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American Football 101

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