Physics of Skydiving

 

The Physics of skydiving involves the interaction between gravity and air resistance. When a skydiver jumps out of a plane, he or she immediately begins to accelerate downwards. As the speed of the falling person increases so does air resistance, so when the skydiver initially begins falling, gravity is stronger than air resistance causing the person to accelerate quickly towards the ground. However, the faster the person falls, the stronger air resistance becomes and eventually the skydiver is moving so fast that the air resistance will equal the force of gravity and the skydiver will no longer accelerate. At this point, the skydiver has reached terminal velocity. Terminal velocity is the maximum velocity attained by a falling object, in this instance a person, as it falls through a fluid, such as air. At terminal velocity the drag force from air resistance balances the force of gravity.  

This free body diagram represents that when drag force is less than the force of gravity acting on the person, m,  to pull them downwards, velocity of the falling person is increasing. However, when drag force is equal to the force of gravity acting on the person, their velocity is equal to terminal velocity and the person is no longer accelerating. 

The drag force is represented as:

The force of gravity is represented as: 

When the skydiver reaches terminal velocity (vt), the drag force is equal to the force of gravity (D = W), represented as: 

If we wanted to solve for a skydiver's terminal velocity we could rearrange the above equation to find: 

For example, if we were to calculate the terminal velocity for an average sized American woman skydiving we could use the above equation and plug in the following values: 

m = 77.56 kg

g = 9.8 m/s^2

C = 1.0 

ρ = 0.9 kg/m^3

A = 0.142 m^2

The mass was taken from the average weight of an American woman. The drag coefficient was taken from a table that listed the typical drag coefficient of a horizontal skydiver as 1.0 (OpenStax). Density of air was found from a table that listed air densities based upon altitude. A typical dive may begin anywhere between 10000 and 15000 feet which corresponds to approximately 3000 to 4500 meters. Thus, a ρ value was averaged from the densities listed for these values and used in the calculation for terminal velocity (“U.S. Standard Atmosphere.”). Finally, using a sketch of a human body estimated by different shapes, the cross sectional area of the skydiver's body was calculated. These calculations can be seen below. 

Using these values and the relationship between the force of gravity and the drag force, a terminal velocity value can be calculated. 

vt = √(2(77.56 kg)(9.8 m/s^2))/ ((1.0)(0.9 kg/m^3)(0.142 m^2))

vt = 103.47 m/s 

This terminal velocity corresponds to roughly 244 mph! However, because the values used in the above equation are estimates and not exact numbers, it is possible that the force of drag was underestimated. Thus, the considerably great terminal velocity that was found may not correspond to actual terminal velocities attained by skydivers in the real world, which are typically around 120 mph. 

Another interesting facet of skydiving is that a falling object with a smaller cross-sectional area, A, will achieve a higher terminal speed than an object with a larger cross-sectional area. Thus, if a skydiver wanted to move more quickly through the air, he or she could do so by decreasing their cross-sectional area by bring their arms in against their body or reorienting their body such that their head was facing down. Similarly, if they wanted to fall more slowly they could make adjustments that would increase their cross-sectional area such as spreading out their arms and legs and falling with their stomach towards the ground in a position called "the spread eagle." 

Lastly, as the skydiver prepares to land they experience a final drag force due to the release of the parachute which slows their descent such that they can land easily and safely. A skydiver will usually deploy their parachute around 6000 feet such that it is open by the time they are 5000 feet above the ground. Once the parachute opens, the cross-sectional area of the falling skydiver is increased and the amount of air resistance he or she encounters is also greatly increased. Air resistance overwhelms the downward force of gravity such that the net force acting on the falling skydiver is upward. An upward net force on a downward falling object causes that object to slow down. Thus, the skydiver will drastically slow down in their descent to the bottom. During their descent, a much slower terminal velocity is attained. 

Works Cited

Skorucak, Anton Skorucak. “What Is the Physics Involved in Skydiving?” PhysLink.com,         www.physlink.com/education/askexperts/ae536.cfm. 

“The Physics Classroom Website.” The Physics Classroom, www.physicsclassroom.com/mmedia/newtlaws/sd.cfm. 

“Physics Of Skydiving.” Real World Physics Problems, www.real-world-physics-problems.com/physics-of-skydiving.html. 

OpenStax. “Physics.” Lumen, courses.lumenlearning.com/physics/chapter/5-2-drag-forces/. 

“U.S. Standard Atmosphere.” Engineering ToolBox, www.engineeringtoolbox.com/standard-atmosphere-d_604.html.