Question

A skydiver of mass \(83.7 \mathrm{~kg}\) (including outfit and equipment) falls in the spread-eagle position, having reached terminal speed. Her drag coefficient is \(0.587,\) and her surface area that is exposed to the air stream is \(1.035 \mathrm{~m}\). How long does it take her to fall a vertical distance of \(296.7 \mathrm{~m} ?\) (The density of air is \(1.14 \mathrm{~kg} / \mathrm{m}^{3}\).)

Short answer

Answer: It takes the skydiver 5.86 seconds to fall a vertical distance of 296.7 m at terminal speed.

Step-by-step solution

Find the terminal speed

To find the terminal speed (v_t), we need to use the formula: $$ v_t = \sqrt{\frac{2m * g}{C * A * \rho}} $$ where m is the mass of the skydiver (83.7 kg), g is the acceleration due to gravity (9.81 m/s²), C is the drag coefficient (0.587), A is the surface area exposed to the air stream (1.035 m²), and ρ is the air density (1.14 kg/m³).

Substitute the given values and find v_t

Now let's plug the given values into the formula: $$ v_t = \sqrt{\frac{2 * 83.7 * 9.81}{0.587 * 1.035 * 1.14}} $$ Evaluating this expression, we find: $$ v_t = \sqrt{\frac{1642.594}{0.6867533}} = 50.6 \thinspace \mathrm{m/s} $$ So, the terminal speed is 50.6 m/s.

Find the time taken to fall the given vertical distance

Now that we know the terminal speed, we can calculate the time taken (t) to fall the vertical distance of 296.7 m, using the formula: $$ t = \frac{s}{v_t} $$ where s is the vertical distance fallen (296.7 m), and v_t is the terminal speed (50.6 m/s).

Substitute the values and find t

Substituting the values into the formula, we get: $$ t = \frac{296.7}{50.6} = 5.86 \thinspace \mathrm{seconds} $$ Hence, it takes the skydiver 5.86 seconds to fall a vertical distance of 296.7 m at terminal speed.

Key concepts

Drag Coefficient

The drag coefficient, denoted as 'C' in physics equations, is a dimensionless number that represents an object's resistance to air or fluid flow. It plays a crucial role when calculating the forces acting on objects moving through a fluid, such as air, which is exactly what happens with a skydiver descending through the atmosphere.

The value of the drag coefficient depends on the shape of the object and the nature of the flow around it. For example, a sleek sports car has a lower drag coefficient than a boxy SUV because the car's design allows air to flow around it more smoothly. Similarly, our skydiver, in a spread-eagle position, increases her surface area while also affecting the drag coefficient compared to a more streamlined, feet-first position.

In practice, engineers and designers work to minimize the drag coefficient to enhance the efficiency of vehicles, aircraft, and even athletic wear for sports to reduce resistance and save energy.

Surface Area

Surface area in the context of a falling object like a skydiver refers to the area exposed to the oncoming air. A larger surface area generally increases the air resistance an object experiences, which impacts the terminal speed it can reach.

When the skydiver falls in a spread-eagle position, she maximizes the surface area, increasing the air resistance. This resistance is beneficial in slowing down descent, allowing for a safer and more controlled fall. By varying her position, she can adjust her fall rate: spread-eagle to slow down or a more streamlined pose to speed up.

Understanding Terminal Speed

In the spread-eagle position, the skydiver reaches a terminal speed where air resistance equals the gravitational pull, resulting in no additional acceleration. Therefore, the increased surface area contributes significantly to reaching terminal speed faster and at a lower velocity.

Air Density

Air density, symbolized by \(\rho\), is the mass per unit volume of air. It changes with altitude, temperature, and humidity, having a considerable effect on the terminal speed of falling objects. At higher altitudes, where air density is lower, objects can reach higher speeds before air resistance forces balance out with gravitational pull.

In calculating the terminal speed of the skydiver, the air density acts opposite to gravity—it's a part of the resisting force. This is why skydivers can fall at different speeds depending on where they are geographically (e.g., sea level vs. high altitude) and under various weather conditions.

In our skydiver’s case, assuming standard conditions and a specified air density of 1.14 kg/m³, we can evaluate the impact of air density on her terminal velocity, as higher air density would result in a lower terminal speed for the same skydiving pose.