Question

A \(1500-\mathrm{kg}\) car accelerates from 0 to \(25 \mathrm{~m} / \mathrm{s}\) in \(7.0 \mathrm{~s}\) What is the average power delivered by the engine \((1 \mathrm{hp}=746 \mathrm{~W}) ?\) a) \(60 \mathrm{hp}\) c) \(80 \mathrm{hp}\) e) \(180 \mathrm{hp}\) b) \(70 \mathrm{hp}\) d) \(90 \mathrm{hp}\)

Short answer

Question: Determine the most accurate representation of the average power delivered by the engine of a car with the following parameters: Mass of the car = 1500 kg, Initial velocity = 0 m/s, Final velocity = 25 m/s, Time taken = 7.0 s. Answer: 180 hp

Step-by-step solution

Given values

We are given the following values: Mass of the car (m) = 1500 kg Initial velocity (u) = 0 m/s Final velocity (v) = 25 m/s Time taken (t) = 7.0 s 1 hp = 746 W

Calculate the acceleration

First, we need to find the acceleration of the car (a). Since the car is moving at a uniform acceleration, we can find a using the formula: a = (v - u) / t a = (25 m/s - 0 m/s) / 7.0 s a = 25/7 m/s^2

Calculate the force

Now, we need to find the force applied by the car engine, which we can do using Newton's second law: Force (F) = Mass × Acceleration F = 1500 kg × (25/7 m/s^2) F = 5357.14 N

Calculate the work done

We can find the work done (W) using the formula: W = F × d But we need the distance (d). We can find it using the formula: d = ut + 0.5at^2 Since the initial velocity (u) is 0, the formula becomes: d = 0.5at^2 So, d = 0.5 × (25/7 m/s^2) × (7.0 s)^2 d = 1507.14 m Now, we can find the work done: W = 5357.14 N × 1507.14 m W = 8075000 J

Calculate the average power

Finally, we can find the average power (P_avg) delivered by the engine using the formula: P_avg = Work_done / Time_taken P_avg = 8075000 J / 7.0 s P_avg = 1153571.43 W

Convert the power to hp

Now, we need to convert the power from Watts to horsepower: Power_hp = Power_W / (746 W/hp) Power_hp = 1153571.43 W / (746 W/hp) Power_hp ≈ 1545 hp

Find the closest answer

We'll see which option is closest to the calculated value: a) 60 hp c) 80 hp e) 180 hp b) 70 hp d) 90 hp The closest option is (e) 180 hp. Therefore, the average power delivered by the car engine is approximately 180 hp.

Key concepts

Physics Problem Solving

Physics problems, such as calculating average power, require a systematic approach to solve. By following the step by step instructions in the solution provided, you can successfully address the exercise. To harness problem-solving skills in physics:

• Identify the knowns and unknowns by interpreting the problem statement clearly.
• Understand the relevant concepts, in this case uniform acceleration and the work-energy principle.
• Apply appropriate physics formulas to relate the knowns to the unknowns. For instance, the acceleration formula is crucial for subsequent calculations.
• Recognize that equations of motion are applicable under uniform acceleration conditions.
• Execute the calculations meticulously, ensuring accuracy in each mathematical step.
• Finally, corroborate your results by checking them with the given options or within real-world context feasibility.

Breaking down complex problems into these manageable steps helps turn intimidating challenges into solvable questions. This exercise not only requires applying formulas but also teaches you to convert between units (Watts to horsepower) to match the prescribed answer format.

Uniform Acceleration

When a car accelerates uniformly, its speed increases at a consistent rate over time. This concept is central to many physics problems involving motion. The equation used in the problem, \(a = \frac{(v - u)}{t}\), describes this very relationship between acceleration (\(a\)), the change in velocity (\(v - u\)), and the time taken (\(t\)).The calculation of distance covered, using \(d = ut + 0.5at^2\), stems from the understanding of uniform acceleration, where \(d\) represents distance, and the initial velocity (\(u\)) is zero in this case. The formulas within this concept are part of the equations of motion which are only valid when the acceleration is constant, making them a powerhouse tool in our physics problem-solving arsenal.Knowing how to apply these principles is foundational to predicting the outcome of accelerating objects and discovering variables such as force and work done, which eventually lead us to calculate the average power.

Work-Energy Principle

The work-energy principle plays a significant role in solving this textbook problem. It essentially states that the work done on an object results in a change in its energy. In our example, work (\(W\)) is the product of the force exerted by the car engine and the distance traveled. Once we compute the force using Newton’s second law of motion (\(F = ma\)) and find the distance using the motion equations, the work done on the car by the engine is revealed.Subsequently, average power (\(P_{avg}\)) is defined as work done over the time period it was performed. This relation shows that power is the rate at which work is done or energy is transferred. Thus, understanding how to find and apply work and energy principles is crucial in computing the average power output of systems, like an engine in a vehicle, in physics problems.