Why do good jumpers seem to hang in the air?

 By Peter J. Brancazio

 Certain athletes are gifted with great jumping ability. We see them most frequently on the basketball court, where the best jumpers' are able to reach well above the 10 foot high rim. Great jumpers also compete in specific track and field events such as the high jump and the long jump. Perhaps the most spectacular jumpers display their skills in a place which is not normally associated with sports or athletics - namely, on the ballet stage. Here, remarkably effortless jumping ability is combined with exceptional body control and grace.

 Great jumpers seem to "hang in the air." They appear to be able to willfully suspend their bodies in flight for what seems like several seconds. Can an athlete really "hang in the air"  in full defiance of the law of gravity or is this a skillfully created illusion? Of course, no person or thing can defy the law of gravity. As we shall see, hanging in the air is an illusion which can be readily explained in terms of basic physics. Indeed, our perceptions of great jumping ability are often hindered by some basic misunderstandings. To begin with, let us consider a simple question: How high can the best jumpers jump?

 Some basketball players are able to touch the top of the backboard, 13 1/2 feet above the floor. However, this is not very remarkable when the player is nearly 7 feet tall and can extend his arms more than three feet above his head. Obviously, the height and reach of the athlete, and not jumping ability alone, determine how high he can reach.

 In the high jump an activity that obviously depends on good jumping ability the athlete takes a few run-up steps, and then launches his or her body over a horizontal bar. The top male high jumpers readily clear 7 feet; the present men's world record is 7'10' ¾” (The current women's high-jump record is 6'9 1/2 ".) However, the technique used in this event is for the jumper to go over the bar with the body roughly parallel to the ground. As a result, a jumper's body is not entirely lifted seven feet above its initial position. The center of the athlete's body may rise vertically 3 to 4 feet during the jump, while the athlete's head may rise only 1 or 2 feet. Here again, the height attained in the jump depends upon the height of the athlete; a taller jumper has an inherent advantage because the center of his or her body is initially higher above the ground (and closer to the bar) than that of a shorter jumper.

 The best measure of jumping ability one that does not depend on the jumper's height  is the standing vertical jump. The individual stands facing a wall, and with arm extended and feet on the floor, makes a mark on the wall at the top of his or her reach. The person jumps vertically, making a second mark at the peak of the jump. The distance between these two marks measures the vertical leap. This is an accurate measure of leaping ability, as each part of the jumper's body rises the same distance. A typical athlete has a vertical leap of 1 1/2 to 2 feet; the best male jumpers attain heights of 3 1/2 to 4 feet.

 The application of the laws of physics to the process of jumping can be very enlightening. The jumper can be thought of as a projectile that must be forcefully launched upwards in order to leave the ground. The jumper creates the launching force by bending into a crouch, then pushing off the ground while straightening the body. The force thus created can exist only as long as the jumper's feet remain in contact with the ground. Once contact is broken, the jumper has nothing to push on to accelerate the body upward, and so immediately begins to slow down in response to the downward pull of gravity. The height of the jump and the "hang time" (the total time the jumper remains airborne) are determined entirely by the jumper's upward speed at the moment the feet leave the ground. From this point on, nothing that the jumper does in the air, such as vigorous pumping of the arms or legs, can extend hang time, for these motions do not retard or counteract the force of gravity.

 The laws governing projectile motion are fairly simple and allow us to relate the height and hang time to the jumper's launching speed. Once any projectile is launched  in this case, when the jumper's feet leave the ground it begins to lose vertical speed at the rate of 32 feet per second (about 22 mph) per second. This projectile continues to rise as it slows until its upward speed has been reduced to zero. At this point, the projectile begins to fall, now gaining speed at exactly the same rate of 32 feet per second. Thus the greater the launching speed, the longer it will take for the upward speed of the projectile to come to zero, and the higher it will rise.

Mathematically, the vertical height (H) of the trajectory is related to the initial upward speed (V) by the equation H = V²/64 (here V must be measured in ft/s). Thus to attain a vertical height of 4 feet, a jumper must attain a vertical speed of 16 ft./sec. (about 11 mi/hr) as he pushes off the ground. The hang time (T) is in turn related to the height H by the equation T= Ö H/2. Thus for a jumping height of 4 feet, the hang time turns out to be one second. This means that the very best jumpers ‑ who rarely exceed a vertical leap of 4 feet  never spend more than one second in the air. To attain a hang time of 2 seconds, a jumper would have to rise vertically 16 feet!

Thus far, we have been considering purely vertical jumping. Now let us look at what happens when the jumper gets a running start, involving a leap which carries the jumper horizontally as well as vertically. One of the basic principles of projectile motion is that the horizontal and vertical components of any motion are independent of each other. In other words, the amount of time it takes for a jumper to rise and fall depends only on the jumper's vertical speed at take-off, and has no relation to the horizontal speed. A running start may help a jumper to gain a bit more vertical speed at takeoff  because the muscles are already in motion and have stored some energy  but by and large it is not possible to convert horizontal motion to vertical motion.

 Finally, the principles of projectile motion can be used to explain the illusion of "hanging in the air". The jumper's vertical speed is greatest at takeoff, and diminishes rapidly to zero as the jumper rises to the peak of the trajectory. Just before and just after reaching the peak, the jumper has very little vertical speed. During this period, the jumper's vertical distance above the ground does not change very much. The projectile ‑ the jumper ‑ will continue to move horizontally at a steady rate while showing very little up or down motion near the peak of the jump. It is this effect that creates the illusion of hanging in the air.

 BIBLIOGRAPHY

 P. Kirkpatrick, Notes on Jumping, Am. J Phys. 25, 614 11957)

 E.L. Offenbacher, Physics and the Vertical Jump, Am. J. Phys. 38, 829 (1970)