Lindsey Franklin
Jason Crasnick
Ryan Connoly
Imagine hurling yourself out of an airplane flying at three hundred miles an hour, ten thousand feet in the air, merely to appease a craving for speed, excitement, and a huge rush of adrenaline. Sounds crazy? Yet, these features make up the extreme sport we call skydiving, which, amazingly, attracts hundreds of people per year. Although the sport appears basic, many aspects of physics are incorporated into each jump. From the initial free-fall of the diver until the moment his or her feet touch the ground, the factors of force, gravity, mass, acceleration, terminal velocity, and air resistance all play an essential role in the divers safety and well-being. In order to evaluate the diver’s jump, we will follow a 75-kilogram female diver for the entirety of her jump. For the sake of simplicity, we will assume that she jumps on a calm, sunny day and is not affected by weather or drastic wind patterns.
The first step in understanding the physics in skydiving is to establish the forces that act on the diver, along with their directions, during every period of her jump. From the moment she leaves the airplane, a number of forces act on the diver. Before she jumps, the forces acting on the airplane act on the diver as well. Due to this fact, the thrust of the airplane pushes her in the direction that the plane is heading, while the drag counteracts the plane’s motion; likewise, the lift of the airplane counteracts the force of gravity. At this particular point in time, the force diagram below represents the forces acting upon the diver and plane:
However, as soon as the diver’s feet leave the platform, the force of gravity pulls her to the ground, as the normal force associated with the airplane disappears. In the lateral direction the force of the air resistance, or drag, acts equally in the opposite direction of the plane as the jump commences, yet the diver is still moving without acceleration in the plane’s direction, as shown in the diagram below (note: there is no force in the direction of the plane due to lack acceleration):
As the jump progresses and the diver has moved far enough out of range to be affected by the drag of the airplane, and the force of drag now acts against gravity. At this point in time, the free-body diagram resembles the following:
This new force of air resistance is affected greatly by surface area. The more surface area a jumper has, the more the air resistance affects the jumper. This difference in surface area is the reason why the parachute is able to slow down the jumper more than his or her own body; the large surface area of the parachute creates more air resistance, causing the diver to fall to the ground more slowly. As the jumper continues downward, the wind, depending on the direction, allowing a variation of the diver’s path, and causing the wind aspect of the freebody diagram of the jumper to fluctuate. However, the force of gravity remains constant, pulling the jumper down with an acceleration of –9.8 m/s^2.
The freebody diagram changes completely when the parachute opens. As the parachute unfastens, the large surface area and unique design both catches and retains wind, and increases the effects of air resistance on the diver. Thus, there is such a drastic change in the velocity of the diver that she is propelled upward, drastically changing the magnitude of the forces acting upon her. Now the force of drag augments enormously due to the increase in surface area, yet still contrasts with the force of gravity. However, the force of gravity continues to pull the diver down after the initial upward movement, due to the disparity in the magnitudes of the forces. Now, the force diagram resembles the following:
While the wind once again plays a major part in the direction of the jumper, the basic forces acting on the diver remain the same until the diver lands safely on the ground.
While determining the direction and identification of forces remains an essential part of the process of comprehending the physics of skydiving, calculating the forces is both imperative and arduous. Because the two main forces acting on the skydiver are the forces of gravity and drag, the equations for both of these forces are quintessential. Newton’s Second law, which states that force is directly proportional to mass and acceleration (F=ma), is used to find the force of gravity. And, because the constant acceleration of gravity is equal to -9.8 m/s^2, the force of gravity can be calculated by multiplying the mass of the diver by the acceleration of gravity constant. For instance, our diver with the mass of 75 kg would have a force of gravity of –9.8m/s^2 x 75kg, or –735 Newtons (because force is a vector quantity, the negative sign indicates the movement in the downward direction).
The air resistance can be calculated by multiplying the atmospheric pressure(the density of the air), velocity (squared), surface area, and the drag coefficient (which is established by the shape of the diver) by one-half (FD=1/2CDr v2A). This is why a diver will fall much faster in a tuck position than if he or she kept a spread eagle position; due to the increase of surface area in the spread eagle position, the air resistance will affect the diver more. Furthermore, as the diver picks up speed the air resistance increases. Likewise, a parachute has more surface area than a spread-eagle position, and thus affects the dive even more.
Now, the magnitudes of the forces described earlier can finally be calculated. In our diver’s first position, the second before she jumped, she was controlled by the forces of gravity, lift, thrust, and drag. Because the airplane is moving at a constant lateral velocity, without any vertical movement (assuming the plane has reached its desired speed and height), it is not accelerating in any direction and thus has a net force of zero. This fact certifies that the forces must be equal and opposite. Thus, we can simply use the equations for gravity and drag to find the magnitudes of the forces for lift and thrust.
To find the force of gravity, we first add the mass of the plane, 3,540 kilograms (the average mass of a small plane), and the mass of our diver, 75 kilograms. The new mass of 3,615 kilograms now is multiplied by the gravity acceleration constant, -9.8 m/s2, resulting in a force of –35427 kgm/s2, or Newtons. Thus, the force of lift is +35427 N. The sidelong movement of drag and thrust is more intricate, due to the perplexities that make up the force of drag. Let the airplane have a velocity of around 119 m/s, (the average speed of a small plane), with a cross-sectional area (the area of the plane that is going against the force of drag, presumably about the size of the cockpit) of four square meters, when air has a density of 1.29 kg/m3. Now, according to the equation of the force of drag, these values are simply multiplied together (after squaring the velocity) along with the coefficient of drag to obtain the force of drag.
For the next sequence of the jump, the force of gravity is obtained the same way, yet this time without the mass of the plane, as the diver is now falling on her own. Thus, the force of gravity is 75 kg multiplied by –9.8m/ s2, or –735 Newtons. The drag, going in the lateral direction, would be her velocity squared, still 14161 m/s (1192), area (around 1.8 meters by .3 meters), and the density of air(1.29 kg/m3) multiplied together. Thus, the force of drag is calculated at 9864.55 multiplied by the coefficient of drag.
For her final segment, the force of gravity remains the same. However, the drag changes drastically, as the force vector has changed direction. Now her surface area is only the area of the bottom of her feet, rather than the front of her body. Her velocity, too, changes, as she is now going downwards.
When discussing skydiving, the speed of the diver is an essential element, as it remains the most sought after characteristic of the sport. Both the lateral and vertical velocities are important. Lateral velocity, however is short-lived: the diver begins with the velocity of the plane, then lateral drag slows down the diver until she maintains little to no lateral movement. The velocity-time graph for the lateral movement of the jumper is resembles the following:
Lateral Velocity versus Time
Vertical motion, however, differs greatly from lateral movement. Instead of leveling off to a lateral velocity of zero, the diver eventually reaches a point of terminal velocity, which can be defined as the fastest one is able to fall in a certain position. All aspects of the dive affect the terminal velocity of a jumper; if surface area, wind, or the design of the parachute varied, the terminal velocity of the diver would vary as well. The diver will eventually reach her terminal velocity when the velocity is so great that the force of air resistance equals the force of gravity. This occurrence signifies that the diver no longer accelerates, and is now traveling with a constant velocity. Thus, the velocity-time graph appears as follows:
Vertical Velocity versus time
The terminal velocity can also be calculated by manipulating both force equations, as it occurs when the two forces equal each other, as they no longer accelerate. Thus, by making the force of gravity equal the force of drag and solving for velocity, one comes up with the equation V=Ö (2mg/CdAr ). If our 75 kg diver had been accelerating at the gravity constant of –9.8m/s2, with a cross-sectional area of .3 square meters, we would find that she has a velocity of 112.52 (Ö 1/CD) m/s. The velocity of the diver remains constant until he/she opens the parachute, which triggers deceleration. Now, the diver will finish his or her dive with the air resistance on the parachute acting as a brake, letting him or her float gently to the ground.
Skydiving is a sport that people of all ages seek to fulfill their desires for speed and excitement. However, it is also a perfect example of how physics is involved in everyday life, and how important it can be regarding the safety and enjoyment of the diver. Using the elements of force, mass, acceleration, gravity, terminal velocity, and air resistance, physics enables us to take part in the intensity of skydiving, as it gives us the knowledge we need to make it safe and fun for all who take part.