Question

A car is traveling at a constant speed of \(26.8 \mathrm{~m} / \mathrm{s}\). It has a drag coefficient \(c_{\mathrm{d}}=0.333\) and a cross-sectional area of \(3.25 \mathrm{~m}^{2} .\) How much power is required just to overcome air resistance and keep the car traveling at this constant speed? Assume the density of air is \(1.15 \mathrm{~kg} / \mathrm{m}^{3}\).

Short answer

Answer: To find the power required, first calculate the air resistance force using the formula \(F_{\text{drag}} = 0.5 * \rho * v^2 * C_{\text{d}} * A\), then calculate the power required using the formula \(P = F_{\text{drag}} * v\). The result will be the power needed to overcome air resistance and keep the car traveling at its constant speed of 26.8 m/s.

Step-by-step solution

Calculate the air resistance force

The air resistance force can be calculated using the following formula: \(F_{\text{drag}} = 0.5 * \rho * v^2 * C_{\text{d}} * A\) Where \(F_{\text{drag}}\) is the air resistance force, \(\rho\) is the density of air, \(v\) is the velocity of the car, \(C_{\text{d}}\) is the drag coefficient, and \(A\) is the cross-sectional area. Using the given values, we can calculate the air resistance force: \(F_{\text{drag}} = 0.5 * 1.15 \mathrm{~kg/m^3} * (26.8 \mathrm{~m/s})^2 * 0.333 * 3.25 \mathrm{~m}^2\) Now, calculate the air resistance force.

Calculate the power required

The power can be calculated using the following formula: \(P = F_{\text{drag}} * v\) Where \(P\) is the power required and \(v\) is the velocity of the car. Using the calculated air resistance force from Step 1 (denoted as \(F_{\text{drag}}\)), we can now calculate the power required: \(P = F_{\text{drag}} * 26.8 \mathrm{~m/s}\) Now, calculate the power required to maintain the car's constant speed while overcoming air resistance. The result will be the power needed to overcome air resistance and keep the car traveling at its constant speed of 26.8 m/s.

Key concepts

Drag Coefficient

The drag coefficient, symbolized by \(C_{\text{d}}\), is a dimensionless number that characterizes the drag or resistance of an object in a fluid environment, such as air or water. It is used to quantify the drag or resistance force that acts opposite to the body's motion through the fluid. The drag coefficient depends on the shape of the object and the flow conditions of the fluid.

The equation to calculate the force of drag on a body moving through air is:\[F_{\text{drag}} = 0.5 \times \rho \times v^2 \times C_{\text{d}} \times A\]where:\begin{itemize}\item \(F_{\text{drag}}\) is the drag force,\item \(\rho\) represents the density of the fluid (air in this case),\item \(v\) is the velocity of the object,\item \(A\) signifies the cross-sectional area of the object facing the fluid flow.\end{itemize}
The value of the drag coefficient is critical in designing vehicles and structures that interact with fluid flow, where low drag is optimal for improving efficiency and reducing energy consumption. In our exercise, a car with a drag coefficient of \(0.333\) indicates that the shape of the car is relatively aerodynamic, contributing to lower energy requirements to overcome air resistance at a given speed.

Power Calculation in Physics

Power in physics is the rate at which work is done or the rate at which energy is transferred. It is an important concept when considering the performance of engines and motors, as well as the physical endurance of moving objects against forces like air resistance. Power is measured in watts (W) in the International System of Units (SI).

The formula to calculate the power required to maintain an object's velocity against the force of drag is given by:\[P = F_{\text{drag}} \times v\]where:\begin{itemize}\item \(P\) is the power,\item \(F_{\text{drag}}\) is the force of air resistance (or drag force),\item \(v\) is the velocity of the object.\end{itemize}
In the context of our exercise, calculating the power required to overcome air resistance involves determining the drag force the car experiences and then multiplying that by the car's constant speed. This result gives us the continuous amount of power the car's engine needs to produce to maintain that speed in the presence of air resistance.

Density of Air

The density of air, denoted by \(\rho\), is the mass per unit volume of air. It is usually measured in kilograms per cubic meter (\(\mathrm{kg/m^3}\)). The density of air is not constant; it varies with altitude, temperature, and humidity. At sea level and at 15 degrees Celsius, the air density is approximately \(1.225 \mathrm{~kg/m^3}\). However, this standard value changes under different conditions. For instance, warmer air is less dense than cooler air.

In calculations involving air resistance, such as the exercise mentioned, the density of air is a crucial factor in determining the drag force experienced by a moving object. The formula for the air resistance force again is:\[F_{\text{drag}} = 0.5 \times \rho \times v^2 \times C_{\text{d}} \times A\]By using the given density of \(1.15 \mathrm{~kg/m^3}\), we acknowledge that the exercise assumes certain atmospheric conditions, likely close to sea level with a mild temperature—conditions that would be common in many real-life situations where such calculations would be relevant for a moving car.