Problem 78 - University Physics with Modern Physics

Question

When it is burned, 1 gallon of gasoline produces \(1.3 \times 10^{8} \mathrm{J}\) of energy. A \(1500-\mathrm{kg}\) car accelerates from rest to 37 \(\mathrm{m} / \mathrm{s}\) in 10 \(\mathrm{s}\) . The engine of this car is only 15\(\%\) efficient (which is typical), meaning that only 15\(\%\) of the energy from the combustion of the gasoline is used to accelerate the car. The rest goes into things like the internal kinetic energy of the engine parts as well as heating of the exhaust air and engine. (a) How many gallons of gasoline does this car use during the acceleration? (b) How many such accelerations will it take to burn up 1 gallon of gas?

Step-by-Step Solution

Answer

(a) The car uses approximately 0.0527 gallons. (b) It takes about 19 accelerations to burn 1 gallon.

Step 1: Calculate the car's kinetic energy (KE)

We first need to calculate the kinetic energy of the car when it reaches the speed of 37 m/s. Use the kinetic energy formula: \( KE = \frac{1}{2}mv^2 \). Substitute \( m = 1500\, \text{kg} \) and \( v = 37\, \text{m/s} \):\[ KE = \frac{1}{2} \times 1500 \times 37^2 = 1,027,500 \, \text{J} \].

Step 2: Calculate the useful energy from the gasoline

With the engine efficiency given as 15\%, the useful energy produced by burning the gasoline to achieve this kinetic energy will be: \( \text{Useful Energy} = \frac{KE}{\text{Efficiency}} = \frac{1,027,500}{0.15} = 6,850,000 \, \text{J} \).

Step 3: Determine gallons of gasoline used for acceleration

The energy released by burning 1 gallon of gasoline is \( 1.3 \times 10^8 \, \text{J} \). To find out how many gallons are used, we solve: \[ \text{Gallons Used} = \frac{\text{Useful Energy Needed}}{\text{Energy per Gallon}} = \frac{6,850,000}{1.3 \times 10^8} \approx 0.0527 \, \text{gallons} \].

Step 4: Calculate number of accelerations per gallon

To find the number of times the car can accelerate using 1 gallon of gasoline, calculate: \[ \text{Number of Accelerations} = \frac{1}{\text{Gallons Used per Acceleration}} = \frac{1}{0.0527} \approx 18.98 \]. Therefore, the car can accelerate approximately 19 times with 1 gallon of gasoline.

Key Concepts

Kinetic Energy Calculation

Understanding the concept of kinetic energy is pivotal in grasping how vehicles move. Kinetic energy is the energy a body possesses due to its motion. For a car accelerating, this energy depends on both its mass and velocity. The formula to calculate kinetic energy is given by: \[ KE = \frac{1}{2}mv^2 \]where \( m \) is the mass of the object and \( v \) is its velocity. - In our example, the car has a mass of 1500 kg and accelerates to a velocity of 37 m/s. - By substituting these values into the formula, we can calculate the car's kinetic energy as 1,027,500 Joules.This energy tells us how much work was required to get the car from a standstill to its operating speed. Being familiar with these calculations not only helps in understanding energy usage and efficiency but also plays a crucial role in designing more energy-efficient automobiles.

Fuel Efficiency

Fuel efficiency is a measure of how effectively a vehicle converts the chemical energy stored in fuel into useful work for motion. This efficiency is often hindered by energy losses in various components of the car, such as:

• Internal friction in the engine and moving parts
• Thermal losses through exhaust and engine heating

For our scenario, the car engine is only 15% efficient. This means only 15% of the fuel's energy is used to accelerate the car, with the remaining energy lost. - The useful energy needed for acceleration is calculated by dividing the kinetic energy by the efficiency. - For this car, achieving a kinetic energy of 1,027,500 Joules requires 6,850,000 Joules of input energy from the gasoline. Understanding fuel efficiency aids in making decisions that are environmentally conscious and economically beneficial, encouraging practices that reduce fuel consumption and emissions.

Physics of Motion

The physics of motion encompasses the principles governing how objects move. In terms of vehicle dynamics, several key physics principles come into play: - **Newton's Laws of Motion:** These laws describe how forces interact with objects to influence motion, such as how a car moves when the engine exerts force on the wheels. - **Acceleration and Velocity:** These terms describe changes in motion. Acceleration refers to the rate of change of velocity that, in this exercise, brings the car from rest to a speed of 37 m/s in 10 seconds. Each of these principles contributes to either the increase or decrease in the vehicle's momentum, thereby affecting its acceleration and deceleration phases. Understanding these dynamics helps one appreciate the complexity involved in designing automobiles that respond predictably and effectively to the forces acting on them.