Solved Problem on Dynamics
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On a windless day, a car travels at a constant speed of 72 km/h, the constant of the car shape c is equal to 0.6 SI units (International System of Units), and the perpendicular area to the direction of motion is 3 m2. Find the magnitude of the drag force.
Problem data:
- Speed of car: v = 72 km/h;
- Constant associated with shape: c = 0.6 SI;
- Cross-section area: A = 3 m2.
Figure 1 shows the elements given in the problem and the air drag force \( {\vec F}_d \) to be calculated.
Solution
First, we convert the car speed given in kilometers per hour (km/h) to meters per second (m/s) used in the International System of Units (S.I.)
\[
\begin{gather}
v=72\;\frac{\mathrm{\cancel{km}}}{\mathrm{\cancel{h}}}\times\frac{1000\;\mathrm m}{1\;\mathrm{\cancel{km}}}\times\frac{1\;\mathrm{\cancel h}}{3600\;\mathrm s}=\frac{72}{3.6}\;\frac{\mathrm m}{\mathrm s}=20\;\mathrm{m/s}
\end{gather}
\]
The magnitude of the air drag force is given by
\[
\begin{gather}
\bbox[#99CCFF,10px]
{F_d=Kv^2}\tag{I}
\end{gather}
\]
where K is the drag coefficient given by
\[
\begin{gather}
\bbox[#99CCFF,10px]
{K=cA}\tag{II}
\end{gather}
\]
substituting the equation (II) into equation (I)
\[
\begin{gather}
F_d=cAv^2\\[5pt]
F_d=\left(0.6\;\mathrm{SI}\right)\left(3\;\mathrm{m^2}\right)\left(20\;\frac{\mathrm m}{\mathrm s}\right)^2
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{F_d=720\;\mathrm N}
\end{gather}
\]
Note: the magnitude of the resistive force is given by
In this problem, the term K was called the drag coefficient, and it depends on another constant c called the shape coefficient
\[
\begin{gather}
F_r=\frac{1}{2}c_r\rho Av²
\end{gather}
\]
where cr is the drag coefficient, ρ is the air density, A is
the cross-sectional area, and v is the speed. The drag coefficient is a dimensionless quantity.In this problem, the term K was called the drag coefficient, and it depends on another constant c called the shape coefficient
\[
\begin{gather}
F_r=\underbrace{\overbrace{\frac{1}{2}c_r\rho}^{c}A}_{K}v²
\end{gather}
\]
In this case, the constant K has dimension of mass per length,
\( \mathrm{M L^{-1}=\frac{kg}{m}} \)
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .