Stagnation Pressure Limit for a Motorcyclist: 04-BS-7, December 2014, Q6

Question 6 — Stagnation Pressure Limit for a Motorcyclist (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Maximum exhale pressure	100 mm water gauge
Density of air (15°C), ρ_air	1.21 kg/m³

Find. The road speed V at which the stagnation pressure at the rider's mouth equals the maximum exhale pressure.

Approach. At a stagnation point the local relative velocity is zero, so by the energy (Bernoulli) equation all of the oncoming air's kinetic energy converts to a local pressure rise (dynamic pressure). Setting this equal to the 100 mm water-gauge exhale limit gives the critical speed.

Convert the exhale limit to a pressure. $$\Delta p = \rho_{water}\, g\, h = 1000\times 9.81\times 0.100 = \boxed{981\text{ Pa}}$$
Equate to the stagnation (dynamic) pressure of the oncoming air and solve for V. $$\tfrac{1}{2}\rho_{air} V^2 = \Delta p \quad\Rightarrow\quad V = \sqrt{\frac{2\Delta p}{\rho_{air}}} = \sqrt{\frac{2(981)}{1.21}} = \boxed{40.3\text{ m/s} = 145\text{ km/h}}$$

Quantity	Result
Pressure equivalent, Δp	981 Pa
Critical speed, V	40.3 m/s (145 km/h)

Topic: Stagnation pressure via the Bernoulli/energy equation.

Key relations: p₀ = p_static + ½ρV² (stagnation pressure at a point where local velocity relative to the flow is zero).

Why this works: at the mouth (a stagnation point relative to the oncoming air), the flow's kinetic energy is fully converted to a local pressure rise the rider physically feels; equating that rise to his known maximum exhale pressure gives the speed at which he can no longer push air out against it.

Common pitfall: using water density in the dynamic-pressure term — the 100 mm water-gauge figure is only a convenient REFERENCE unit for the pressure itself; the fluid actually hitting the rider is air, so ρ_air belongs in ½ρV².

Source: White, Fluid Mechanics, 8th ed., §3.7 (Bernoulli equation; stagnation pressure and the Pitot-tube principle).

Question 6 of 13: Stagnation Pressure Limit for a Motorcyclist

Question 6 — Stagnation Pressure Limit for a Motorcyclist (5 marks)