EXAM · FRICTION AND DRAG
TERMINAL VELOCITY QUADRATIC DRAG
A skydiver of mass m = 75 kg falls belly-down through air (density ρ = 1.225 kg/m³) with cross-sectional area A = 0.7 m² and drag coefficient C_d = 1.0. Using the Newtonian quadratic drag law F_drag = ½ρC_dAv², find the quadratic drag constant k, the gravitational force, and the terminal velocity the skydiver approaches.
Linked equations:F_d = \tfrac{1}{2}\rho C_d A v^2v_t = \frac{mg}{b}\text{ (linear)},\quad v_t = \sqrt{\frac{2mg}{\rho C_d A}}\text{ (quadratic)}
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Step-by-step solution
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Step 1
What is the quadratic drag constant k = ½ρC_dA for this skydiver?
Hint
k combines all the non-velocity factors in the quadratic drag law: k = ½ · ρ · C_d · A. The drag force is then F = k · v².
Step 2
What is the gravitational force on the skydiver?
Step 3
At terminal velocity, drag equals gravity. Solve for v_t.
Solution walkthrough
Newtonian drag is the dominant regime whenever a body moves fast through a fluid. The drag force F = ½ρC_dAv² depends on the square of speed — doubling speed quadruples drag. Collecting the constant factors: k = ½ × 1.225 × 1.0 × 0.7 = 0.4288 kg/m. This k is the 'drag coefficient per unit speed squared'.
Gravity pulls the skydiver down with F_grav = 75 × 9.807 = 735.5 N. As the skydiver accelerates, drag increases as v². Terminal velocity is where the two forces balance: k·v_t² = F_grav, so v_t = √(F_grav/k) = √(735.5/0.4288) ≈ 41.4 m/s.
The square-root dependence is important: doubling mass increases v_t by only √2 ≈ 1.41. Doubling drag area A halves v_t² and reduces v_t by √2 ≈ 0.71. This is why a skydiver in a head-down dive (smaller A, lower C_d) can reach 90 m/s or more, while a fully spread parachute with enormous A and C_d ≈ 1.5 cuts v_t to around 5 m/s — a safe landing speed.
The approach to v_t under quadratic drag is not a simple exponential (that was the linear-drag case). It follows a hyperbolic tangent: v(t) = v_t · tanh(t / τ_q) where τ_q = m/(k·v_t). The shape is qualitatively similar — rapid rise then flattening — but the math is richer.
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