Problem 67 - College Physics

Question

A race car is driven at a constant velocity of \(200 \mathrm{~km} / \mathrm{h}\) on a straight, level track. The power delivered to the wheels is \(150 \mathrm{~kW}\). What is the total resistive force on the car?

Step-by-Step Solution

Answer

The total resistive force on the car is approximately 2700 N.

Step 1: Convert Velocity Units

First, convert the car's velocity from kilometers per hour (km/h) to meters per second (m/s) to work with the power in watts. Use the conversion factor where 1 km/h equals approximately 0.27778 m/s: \(200 \text{ km/h} \approx 200 \times 0.27778 \text{ m/s} = 55.556 \text{ m/s}\).

Step 2: Write the Power Formula

The power delivered to the car's wheels is related to the force and velocity by the formula: \(P = F \times v\), where \(P\) is power, \(F\) is force, and \(v\) is velocity.

Step 3: Rearrange to Solve for Force

Rearrange the power formula to solve for force: \(F = \frac{P}{v}\). This allows us to calculate the resistive force exerted on the car.

Step 4: Substitute the Given Values

Substitute in the given values: \(P = 150,000 \text{ W} = 150 \text{ kW}\) and \(v = 55.556 \text{ m/s}\). Thus, \[F = \frac{150,000}{55.556} \approx 2700 \, \text{N}\].

Step 5: Interpret the Result

The calculated force \(2700 \text{ N}\) represents the total resistive force acting against the motion of the car on the track, which includes air resistance, friction, and any other opposing forces.

Key Concepts

Kinematics

Kinematics deals with the motion of objects without taking into account the forces that cause that motion. It focuses on describing how objects move, using parameters such as velocity, speed, and time.
In this exercise, kinematics is essential to understand how the car moves on the track. The car is traveling at a constant velocity of 200 km/h, which we convert to meters per second (m/s) - an essential step for calculating power and resistive forces accurately.
Understanding this conversion is vital:

• Velocity in km/h needs to be converted to m/s for compatibility with standard physics equations.
• Converting 200 km/h results in approximately 55.556 m/s.

This velocity tells us about the rate of motion but not about the forces affecting the car's motion, which we'll address in further sections.

Power and Energy

Power and energy are closely related concepts in physics—power being the rate at which energy is transferred or converted. In the given problem, power is delivered to the car's wheels to maintain motion.
The power here is given as 150 kW, which can be converted to watts (W) for use in calculations. One of the key formulas involving power is:

• Power equals force multiplied by velocity: \(P = F \times v\).
• This equation can help us understand how much work is done by the car's engine every second.
• By knowing the power and velocity, you can determine the force exerted, which directly impacts what force is required to overcome resistive forces.

Reworking this formula allows us to calculate the unknown resistive force acting on the car, using the known values of power and velocity.

Resistive Forces

Resistive forces are forces that oppose the motion of an object. These could include friction, air resistance, and other factors. In our scenario, they are added together as a single force that opposes the movement of the car.
In the exercise, we've calculated the resistive force to be approximately 2700 N. This force is essential in determining the net effect of the surrounding environment on the car:

• Resistive forces work against the engine's generated power, effectively slowing down the car if not countered.
• These forces ensure that the car must maintain a certain power output to keep moving at constant speed.
• Understanding resistive forces helps in optimizing vehicle efficiency, by finding ways to reduce these forces in practical scenarios.

By calculating these forces, we can better judge how much power is necessary for a car to maintain a desired speed, considering environmental resistances.