A 1400 Kg Sports Car Accelerates From Rest To 95.0 Km/h In 7.4 Seconds. How To Calculate Average Power Delivered By The Engine?

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When we delve into the world of physics, analyzing the motion and power of vehicles, especially sports cars, presents a captivating challenge. Consider, for example, a 1400 kg sports car that accelerates from a standstill to 95.0 km/h in a mere 7.4 seconds. This scenario opens up a plethora of questions about the car's performance, the engine's capabilities, and the underlying physics principles at play. Specifically, one might ask: What is the average power delivered by the engine during this impressive acceleration? To unravel this, we must embark on a journey through kinematics and dynamics, exploring concepts such as acceleration, velocity, and the fundamental relationship between power, work, and energy.

First, let's break down the problem. We know the car's mass (1400 kg), its initial velocity (0 km/h, since it starts from rest), its final velocity (95.0 km/h), and the time it takes to reach this velocity (7.4 seconds). The key to finding the average power lies in understanding the relationship between power, work, and energy. Power, in its simplest form, is the rate at which work is done, or the rate at which energy is transferred. In this case, the engine's power is being used to increase the car's kinetic energy, which is the energy an object possesses due to its motion. The formula for kinetic energy is given by KE = 1/2 * m * v^2, where m is the mass and v is the velocity.

Before we can calculate the kinetic energy, we need to ensure all our units are consistent. The velocity is given in kilometers per hour (km/h), but the standard unit for velocity in physics calculations is meters per second (m/s). Therefore, we must convert 95.0 km/h to m/s. To do this, we multiply by 1000 (since there are 1000 meters in a kilometer) and divide by 3600 (since there are 3600 seconds in an hour). This conversion yields a final velocity of approximately 26.39 m/s. Now we have all the necessary information to calculate the change in kinetic energy, which represents the work done by the engine.

Calculating Average Acceleration

The initial step in determining the average power delivered by the engine involves calculating the average acceleration of the sports car. Average acceleration is defined as the change in velocity over a period of time. To compute average acceleration, we use the formula:

a = (v_f - v_i) / t

Where:

  • a represents the average acceleration.
  • v_f is the final velocity.
  • v_i denotes the initial velocity.
  • t signifies the time interval.

In the context of our 1400 kg sports car, which accelerates from rest (v_i = 0 m/s) to 95.0 km/h (26.39 m/s) in 7.4 seconds, we can substitute these values into the formula to find the average acceleration. By plugging in the numbers, we get:

a = (26.39 m/s - 0 m/s) / 7.4 s

This calculation yields an average acceleration of approximately 3.57 m/s^2. This value is crucial because it helps us understand how the car's velocity changes over time during the acceleration phase. The average acceleration provides a constant rate at which the car's speed increases, allowing us to quantify the rate of change in motion. This parameter is not only essential for determining the forces acting on the car but also for calculating the power required to achieve such acceleration.

Understanding the average acceleration is pivotal for several reasons. First, it gives us a clear picture of the car's performance capabilities. An acceleration of 3.57 m/s^2 indicates a strong and rapid increase in speed, which is characteristic of a sports car. Second, the average acceleration is a key component in calculating the force required to accelerate the car, according to Newton's second law of motion (F = ma), where F is force, m is mass, and a is acceleration. Knowing the force, in turn, allows us to calculate the work done by the engine, which is the force applied over a distance. This work is directly related to the power delivered by the engine, as power is the rate at which work is done.

Moreover, the average acceleration provides a basis for comparison with other vehicles. Sports cars are often characterized by their ability to accelerate quickly, and a high average acceleration is a hallmark of a powerful engine and efficient vehicle dynamics. By understanding this parameter, we can quantitatively compare the performance of different vehicles and appreciate the engineering that goes into designing a high-performance sports car. The calculation of average acceleration is thus a fundamental step in analyzing the motion and power dynamics of the sports car, paving the way for a deeper understanding of the energy transfer and engine capabilities involved in the acceleration process.

Calculating the Work Done and Average Power

Having determined the average acceleration, the subsequent step involves calculating the work done by the engine, which is crucial for finding the average power. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. Therefore, by calculating the change in kinetic energy of the sports car, we can determine the work done by the engine during the acceleration phase. The formula for kinetic energy (KE) is given by:

KE = 1/2 * m * v^2

Where:

  • KE represents the kinetic energy.
  • m is the mass of the object.
  • v is the velocity of the object.

In our scenario, the car starts from rest, so its initial kinetic energy (KE_i) is zero. The final kinetic energy (KE_f) is calculated using the final velocity of 26.39 m/s and the mass of 1400 kg. Plugging these values into the formula, we get:

KE_f = 1/2 * 1400 kg * (26.39 m/s)^2

This calculation yields a final kinetic energy of approximately 487,147 Joules (J). Since the initial kinetic energy was zero, the change in kinetic energy, and thus the work done by the engine, is equal to this final kinetic energy. Therefore, the engine does 487,147 J of work to accelerate the car to 95.0 km/h.

With the work done known, we can now calculate the average power delivered by the engine. Power is defined as the rate at which work is done, or the work done per unit of time. The formula for average power (P_avg) is:

P_avg = W / t

Where:

  • P_avg represents the average power.
  • W is the work done.
  • t is the time interval.

Substituting the work done (487,147 J) and the time interval (7.4 seconds) into the formula, we get:

P_avg = 487,147 J / 7.4 s

This calculation results in an average power of approximately 65,831 Watts (W). To express this power in a more common unit for engines, we can convert Watts to horsepower (hp), using the conversion factor 1 hp = 746 W. Dividing the power in Watts by 746, we get:

P_avg (hp) = 65,831 W / 746 W/hp

This gives us an average power of approximately 88.2 horsepower. This value indicates the rate at which the engine is performing work to accelerate the car, providing a measure of the engine's capability during this specific acceleration phase. The average power delivered by the engine is a critical parameter in assessing the car's performance, as it reflects the engine's ability to convert fuel energy into mechanical work efficiently and rapidly.

Practical Implications and Considerations

The calculated average power of 88.2 horsepower provides valuable insights into the performance of the sports car's engine. However, it is essential to understand the practical implications and consider the various factors that influence this value. The average power calculated represents the engine's output during the 7.4-second acceleration phase. It does not represent the engine's peak power, which is the maximum power the engine can produce under ideal conditions. The peak power of a sports car engine is typically higher than the average power used during a specific acceleration, as the engine is designed to have reserve capacity for bursts of speed and demanding driving conditions.

Several factors can affect the actual power delivered by the engine in real-world scenarios. One significant factor is mechanical efficiency. The engine's mechanical efficiency refers to how effectively it converts the chemical energy of the fuel into mechanical work. In a real engine, not all the energy from the fuel is converted into useful work; some energy is lost due to friction, heat, and other inefficiencies. This means that the actual power delivered to the wheels may be less than the theoretical power calculated based on the change in kinetic energy. The drivetrain, which includes the transmission, driveshaft, and differential, also experiences frictional losses, further reducing the power available at the wheels.

Another factor to consider is air resistance. As the car accelerates, it encounters increasing air resistance, which opposes its motion. The engine must expend additional power to overcome this resistance, which means that some of the power output is used to combat air drag rather than to increase the car's speed. At higher speeds, air resistance becomes a significant factor, requiring a substantial portion of the engine's power output. This is why the power required to maintain a constant high speed is much greater than the power required to accelerate from rest to a moderate speed.

The road conditions and tire grip also play a crucial role in the car's acceleration and the power delivered by the engine. If the road surface is slippery or the tires lack sufficient grip, some of the engine's power may be wasted in wheel spin rather than being effectively transferred to the road. This reduces the car's acceleration and the overall efficiency of the power delivery. Similarly, uphill or downhill gradients will affect the power required for acceleration. Accelerating uphill requires more power to overcome gravity, while accelerating downhill may require less power.

Additionally, the engine's power output can vary depending on the engine's operating conditions, such as temperature, air density, and fuel quality. Engines are typically optimized to deliver peak power within a specific range of engine speeds (revolutions per minute, or RPM). The power output may be lower at very low or very high RPMs. The driver's gear selection and throttle input also influence the engine's power delivery. A skilled driver can optimize gear changes and throttle control to maximize the car's acceleration and overall performance.

Conclusion

In conclusion, determining the average power delivered by the engine of a 1400 kg sports car accelerating from rest to 95.0 km/h in 7.4 seconds involves a series of calculations rooted in the principles of physics. By calculating the average acceleration, the change in kinetic energy, and the work done, we arrived at an average power output of approximately 88.2 horsepower. This value provides a quantitative measure of the engine's performance during this specific acceleration phase. However, it is crucial to recognize that this is an average value and that various factors, such as mechanical efficiency, air resistance, road conditions, and engine operating conditions, can influence the actual power delivered in real-world scenarios.

Understanding these concepts not only enhances our appreciation for the engineering and physics behind high-performance vehicles but also provides a framework for analyzing and comparing the performance characteristics of different vehicles. The interplay between power, acceleration, and the various factors that affect them highlights the complexity and sophistication of automotive engineering. By applying fundamental physics principles, we can gain valuable insights into the dynamics of motion and the capabilities of powerful machines like sports cars.