Mathematics of skydiving
The mathematics of skydiving encompasses various principles of physics and fluid dynamics, focusing on the forces acting on a skydiver during free fall and parachute deployment. When a skydiver jumps from an airplane, they experience acceleration due to gravity until they reach terminal velocity—the point where gravitational force equals drag force, resulting in zero acceleration. This terminal velocity can vary based on body position, with a belly-to-earth orientation leading to speeds around 122 mph, while diving head-first can increase speeds to nearly 200 mph.
The performance of the parachute is critical and is influenced by the canopy's wing-load, determined by the jumper's weight and equipment. The relationship between weight, drag, and descent rates is described by differential equations that model the skydiver's changing velocity and position throughout the jump. A parachute, once deployed, significantly reduces terminal velocity to safer levels, typically between five and ten mph, by increasing cross-sectional area and drag coefficient. Understanding these mathematical and physical concepts enhances both the safety and enjoyment of the skydiving experience, making it an intriguing intersection of mathematics, physics, and adventure.
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Mathematics of skydiving
Summary: Principles of calculus can be used to model a sky dive and to calculate the effect of the parachute on velocity.
Skydiving is the act of leaping out of an airplane at a sufficient altitude and placing your life in the hands of a piece of cloth—although a fairly large piece of cloth. Leonardo da Vinci left drawings of parachutists in his Codex Atlanticus circa 1485. The modern parachute was invented by Louis-Sébastien Lenormand in France, making the first public jump in 1783. In 1797, André Garnerin was the first to use a silk parachute, earlier versions being made of linen. The first parachute jump from an airplane was in Venice Beach, California, in 1911. The parachute was held in the arms and thrown out as the jumper left the plane. The soft-pack parachute was developed in 1924. There are two types of parachutes used for skydiving: round, and ram-air (square). The U.S. Army uses the round 35-foot diameter parachute to train its paratroopers because they are reliable and give the jumper a terminal velocity of about 15 feet per second. Most skydivers in the United States started using a 28-foot round canopy. They produced a terminal velocity of about 17–18 feet per second—a somewhat hard landing. The switch to ram-air types came in the 1970s; these give more comfortable landings and maneuverability. The rates of descent vary from canopy to canopy, but terminal velocities usually run from eight feet per second (5.5 mph) to 14 feet per second (9.5 mph).
A canopy’s performance is determined by its wing-load, which helps determine the terminal velocity and speed at landing. Most canopies are flown with a wing-load between 0.8 and 2.8 pounds per square foot. To compute the right size of canopy, take the total weight (W) of the jumper and equipment divided by the assigned wing-load factor (WLF):
To model the parachute jump itself is much more complicated. It involves a first order differential equation to find the speed. The forces on a skydiver are the gravitational force, Fg, and the drag force, Fd, of air resistance and buoyancy. There are two factors to the drag: the time before and the time after the canopy deploys. If x is the distance above the Earth’s surface, then a=dv/dt is acceleration and v=dx/dt is velocity. For most jumps, the gravitational force stays essentially constant.
In a first approximation to the problem, take the drag force to be proportional to the velocity. The coefficient of drag has one value when the skydiver is falling and a second value when the parachute is fully deployed. During the fall, the velocity satisfies the initial value problem:
This is a separable ordinary differential equation. Its solution can be found by most students in a calculus class. The jumper’s position then is found by integrating the velocity with initial condition that at time t=0 the jumper is at the jump altitude. After the chute deploys, the velocity and position can be found exactly as above, except that the drag coefficient and initial conditions change.
A second approach is to assume that the drag force is proportional to the square of the speed. Then, a falling object reaches a terminal velocity:
where VT is the terminal velocity, m is the mass of the falling object, g is the acceleration due to gravity, Cd is the drag coefficient, ρ is the density of the fluid through which the object is falling, and A is the projected area of the object.
Based on air resistance, the terminal velocity of a skydiver in a belly-to-Earth free-fall position is about 122 miles per hour (179 feet per second). A jumper reaches 50% of terminal velocity after about three seconds and reaches 99% in about 15 seconds. Skydivers reach higher speeds by pulling in limbs and flying head down, reaching speeds close to 200 miles per hour. The parachute reduces the terminal velocity to the five to 10 miles per hour range. This is achieved by increasing the cross-sectional area and the drag coefficient, lowering the terminal speed.
Bibliography
Meade, Doug. “Maple and the Parachute Problem: Modeling With an Impact.” MapleTech 4, no. 1 (1997).
Meade, Doug, and Allan Struthers. “Differential Equations in the New Millennium: The Parachute Problem.” International Journal of Engineering Education 15, no. 6 (1999).
Poynter, Dan, and Mike Turoff. Parachuting. 10th ed. Santa Barbara, CA: Para Publishing, 2007.
The United States Parachute Association. http://www.uspa.org/.