Terminal Velocity

#362 / Fluids Medium

Find the terminal velocity of a 70 kg skydiver falling in a spread eagle position if their 1.1 m^2 projected area has an associated drag coefficient of 1.5. Assume the density of air is 1.22 kg/m^3.

Expand Hint

The drag force on objects immersed in a large body of flowing fluid or objects moving through a stagnant fluid:

$$$D=\frac{1}{2}\rho U^{2}C_{D}A$$$

where $$C_D$$ is the drag coefficient, $$U$$ is the flowing fluid or moving object’s velocity, $$\rho$$ is the fluid density, and $$A$$ is the projected area of blunt objects with axes perpendicular to the flow.

Hint 2

$$$Force=mass \times acceleration$$$

The drag force on objects immersed in a large body of flowing fluid or objects moving through a stagnant fluid:

$$$D_f=\frac{1}{2}\rho U^{2}C_{D}A$$$

At terminal velocity, the net force is zero, meaning the drag force on the skydiver is equal to the force of gravity. Remember, $$Force=mass \times acceleration$$ , so

$$$F_{weight}=mg=D_f$$$

where $$m$$ is the mass and $$g$$ is the acceleration due to gravity.

Thus,

$$$mg=\frac{1}{2}\rho U^{2}C_{D}A$$$

Solving for terminal velocity:

$$$U=\sqrt{\frac{2mg}{\rho C_DA}}=\sqrt{\frac{2(70kg)(9.8m)(m^3)}{(s^2)(1.22kg)(1.5)(1.1m^2)}}$$$

$$$=\sqrt{\frac{1372m^2}{2.013s^2}}=\sqrt{681.57m^2/s^2}=26.1\:m/s$$$

26.1 m/s

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