Brainteaser Physics: Challenging Physics Puzzlers

5. Not Exact but Still Relevant

? ≥ Ω + √ > cm ? π 0). Does it matter? Take the realistic values L = 75 m, u = 1.5 m/s, and v = 1 m/s. That gives t = 100 s for walking on the corridor floor, and 180 s for walking on the belt, that is, a very significant difference. To be more precise, let us calculate how much faster you must walk, compared with your friend, if you both return at the same time. Your speed is still u but your friend’s speed on the corridor floor is w. You arrive back simultaneously if which can be written as A very fast walk corresponds to, say, u = 2 m/s. If 2 m/s is your speed and if the speed of the belt is still v = 1 m/s, your friend can walk on the corridor floor at the more modest pace of 1 m/s without returning later than you. Note that the time t depends on the square of v. A “negative” w u v u = − 1 2 2 2 2 1 2 2 L w L u v u = ⋅ − 1 / t L u v L u v Lu u v L u v u = + + − = − = ⋅ − 2 2 1 2 2 2 2 1 / 112 brainteaser physics speed would give the same time. Thus, it does not matter if you start walking in the same direction as the belt is moving and then return on the belt, or if you do it in the reverse order. Our problem has a close analogy in one of the most famous experiments in physics—the Michelson–Morley experiment in 1887 to determine whether the speed of light depends on the speed of the light source through an assumed stationary ether. 5.3 Shot Put and Pole Vault For the female athletes the shot passes approximately at the level of the bar in pole vaulting. For male athletes the shot does not quite reach the same height as the bar. Recall a well-known result for the range s of a parabolic flight path, when the projectile is launched with speed v at an angle a to the horizontal ground. It is where g is the acceleration of gravity. The maximum range s is obtained for a = 45°. Then the maximum height of the trajectory is found to be The best female and male shot putters reach about 21 m (69 ft). The shot does not leave from the ground level, however, but from a hand that is about 2 m higher up. If we still imagine that s corresponds to a trajectory starting at the ground, and at an angle a = 45°, the shot has already covered a horizontal distance of about 2 m when it leaves the hand. That makes s longer by about 2 m than the result recorded at the competition. The expression h = s/4 then suggests that the shot passes almost 6 m above the ground. This is also about the height reached by the male pole-vaulter, but it is about 1.5 m higher than the result for the female pole-vaulter. Considering the uncerh s = 4 s v g = 2 2 sin( ) a not exact but still relevant 113 tainties in our model it is best to conclude that the maximum height of the shot is somewhat lower than the level of the bar in pole vaulting for men, and about the same as that level for women. The winners ’ results at the 2004 Olympic Games in Athens were: pole vault, 4.91 m for women and 5.95 m for men; shot put, 21.06 m for women (shot mass, 4.00 kg) and 21.16 m for men (shot mass, 7.26 kg). Any caveat? The release angle a affects the possible force on the shot, and therefore also its speed v. Since the aim is to maximize the actual s(a), we have ds(a)/da at the optimum a. This gives only a small (second-order) effect in s but affects the height in a complicated way. Many athletes let a be about 35°. Because h/s = 1/4 tan a we have overestimated the height of the shot if we let a = 45°. The distance between the bar and the center of mass of the polevaulter is very small when the bar is passed. In fact, the center of mass often passes slightly below the bar, as...