## Drag Force

What factor does the drag force increase by as a car travels from 50 to 100 km/hr?

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 The only variables changing between the two speeds is the drag force and velocity. All other factors remain constant. $$\frac{D_1}{U_1^2}=\frac{1}{2}\rho C_{D}A=\frac{D_2}{U_2^2}$$$
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. We can write two equations to solve for the drag force at each speed: $$D_1=\frac{1}{2}\rho U_1^{2}C_{D}A$$$
$$D_2=\frac{1}{2}\rho U_2^{2}C_{D}A$$$The only variables changing between the two speeds is the drag force and velocity. All other factors remain constant. We can set the constants from the two equations equal to each other to create a proportional relationship: $$\frac{D_1}{U_1^2}=\frac{1}{2}\rho C_{D}A=\frac{D_2}{U_2^2}$$$
Thus, the drag force is increasing by a factor of:
$$\frac{D_2}{D_1}=\frac{U_2^2}{U_1^2}=\frac{(100km/hr)^2}{(50km/hr)^2}=\frac{10,000}{2,500}=4$$\$
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