BIEN 301 Lopez P 7 59 Rosalyn Pillow

BIEN 301 Lopez P 7. 59 Rosalyn Pillow February 1, 2007

Joe can pedal his bike at 10 m/s on a straight level road with no wind. The rolling resistance of his bike is 0. 80 N-s/m. The drag area (CDA) of Joe and his bike is 0. 422 m 2. Joe’s mass is 80 kg and that of the bike is 15 kg. He now encounters a head wind of 5. 0 m/s.

(A) Develop an equation for the speed at which Joe can pedal into the wind. [Hint: A cubic equation for V will result. ] (B) Solve for V, that is how fast can Joe ride into the headwind (C) Why is the result not 10 – 5 = 5 m/s as one might first suspect

n n n n Cr = 0. 8 N-s/m Vw = 5 m/s Vj = 10 m/s CDA = 0. 422 m 2 mj = 80 kg mb = 15 kg CD = drag/(½ρV 2 A)

• • Perfectly symmetrical – pedal asymmetry negligible → CD = drag/½V 2 A, no moment Road remains smooth and level Joe exerts same power in both scenarios, thereby mathematically relating them Other drag variables are negligible – lift, pressure differences, etc

Drag = ½CDAρV 2 (Equation 1) Resistance = Cr(V) (Equation 2) ΣF = Drag + Resistance (Equation 3) Po = P 2 (Equation 4) P = FV (Equation 5) V = Vj + Vw every time in calculating the drag

Fr 1 = R*Vo = 0. 8 N-s/m*10 m/s = 8 N Fjoe, bike 1 = ½ρ CDA V 2 = ½(0. 422 m 2)(1. 2 m 3)(10 m/s)2 = 25. 32 N Fo = Fr + Fjoe, bike 1 = 33. 32 N Po = Fo. Vo = (33. 32 N)(10 m/s) = 333. 2 J/s

Calculations Continued F 1 F 1 P 1 = ½Cd. Aρ(Vj + Vw)2 + Cr(Vj +Vw) = ½(0. 422 m 2)(1. 2 kg/m 3)(Vj + 5)2 + (0. 8 N-s/m)(Vj + 5) = [0. 2532(Vj 2 + 10 Vj + 25) + 0. 8 Vj + 4] N = F 1 V 1 = 0. 2532 Vj 3 + 3. 332 Vj 2 + 10. 33 Vj

Calculations part 3 Po = P 1 333. 2 = F 1 V 1 333. 2 = Vj[0. 2532(Vj 2 + 10 Vj + 25) + 0. 8 Vj + 4] 0 = 0. 2532 Vj 3 + 3. 332 Vj 2 + 10. 33 Vj - 333. 2 Vj = 7. 1131993 m/s

Why NOT? NOT 10 – 5 = 5 m/s = V n both velocities cause air to flow over the drag area in the same direction, n added then squared in the drag equation, and n the wheel resistance which is linear must also be measured and included

This type of fluid mechanics involving drag, density, area and resistance on a macro level, can be applied to n Estimate the time required for a medicine/radioactive isotope/nanoparticle to travel a specified distance through the body in and against various fluids.