Problem 15 - An Introduction to Mechanics

Question

A 1800 -lb sportscar accelerates to \(60 \mathrm{mi} / \mathrm{h}\) in \(4 \mathrm{~s}\). What is the average power that the engine delivers to the car's motion during this period? For consistency, we are using the definition \(1 \mathrm{hp}=746 \mathrm{~W}\).

Step-by-Step Solution

Answer

The engine's average power is approximately 99.1 hp.

Step 1: Convert units from miles per hour to feet per second

To start, we need to convert the car's velocity from miles per hour to feet per second, since power calculations usually involve feet and seconds. The car accelerates from 0 to 60 mph. 1 mile = 5280 feet and 1 hour = 3600 seconds, thus:\[60 \text{ mph} = 60 \times \frac{5280}{3600} \text{ ft/s} \approx 88 \text{ ft/s}\]

Step 2: Calculate the work done in accelerating the car

The work done on the car can be found using the formula for kinetic energy, \( W = \Delta KE \). Since the car starts from rest, the initial kinetic energy is 0. The final kinetic energy is:\[ KE = \frac{1}{2} mv^2 \]The car's weight (force due to gravity) is 1800 lb. Using \( g = 32 \text{ ft/s}^2 \), we find mass \( m \) as follows: \[ m = \frac{\text{weight}}{g} = \frac{1800}{32} \approx 56.25 \text{ slugs} \]Substitute to find work done:\[ W = \frac{1}{2} \times 56.25 \times (88)^2 \approx 217800 \, \text{ft-lb} \]

Step 3: Determine the power required

Power is the rate of doing work, calculated by the work done divided by the time taken. The work calculated is 217800 ft-lb and the time taken is 4 seconds:\[ P = \frac{W}{t} = \frac{217800}{4} = 54450 \text{ ft-lb/s} \]Power in ft-lb/s needs to be converted to watts. Since 1 hp = 746 W and 1 hp = 550 ft-lb/s, use the conversion:\[ P = \frac{54450 \text{ ft-lb/s}}{550} \times 746 \approx 73926.73 \text{ W} \]

Step 4: Convert power to horsepower

Finally, convert the power in watts to horsepower using the given conversion factor:\[P_{\text{hp}} = \frac{73926.73}{746} \approx 99.1 \, \text{hp}\]

Key Concepts

Understanding Kinetic Energy

Kinetic energy is a fundamental concept in physics that quantifies the energy an object possesses due to its motion. Whenever an object is moving, it has kinetic energy. It depends on two main factors:

• Mass of the object (m)
• Velocity of the object (v)

The formula to calculate kinetic energy is:\[KE = \frac{1}{2} mv^2\]This formula tells us that the kinetic energy increases with the square of the velocity, meaning if you double the speed of an object, its kinetic energy increases by four times.
In our exercise, the sportscar starts from rest, so its initial kinetic energy is zero. As it accelerates, its speed increases, allowing us to calculate the change in kinetic energy, which gives us the work done on the car. Using the mass converted from weight and velocity in feet per second, we calculate the final kinetic energy.

Navigating Unit Conversion

Unit conversion is an essential skill in physics, allowing for calculations using consistent units. It ensures accuracy in computations by converting units like miles per hour to feet per second. To convert miles per hour (mph) to feet per second (ft/s), we use the conversion factors:

• 1 mile = 5280 feet
• 1 hour = 3600 seconds

Using these, we convert the velocity of 60 mph to feet per second for the sportscar:\[60 \text{ mph} = 60 \times \frac{5280}{3600} \text{ ft/s} = 88 \text{ ft/s}\]This conversion is crucial as power and energy calculations typically require units of feet and seconds, especially when working with kinetic energy or force, commonly measured in pounds-force (lb). In physics problems, matching units across different measurements simplifies the problem-solving process, reducing errors and increasing clarity.

Decoding Horsepower

Horsepower is a unit of power used to describe the rate at which work is done, and it is commonly used for measuring the power output of engines. Understanding how to work with horsepower involves recognizing its relationship with other units of power such as watts and feet-pounds per second.

• 1 horsepower (hp) = 746 watts (W)
• 1 horsepower (hp) = 550 foot-pounds per second (ft-lb/s)

To determine the engine's power output in the problem, we start by calculating the work done and then the power in ft-lb/s. Using unit conversion, we express this power in watts first, then convert to horsepower with the given factor.
Converting power to horsepower effectively communicates engine efficiency and capability to an audience, particularly when summarizing performance specifications in the automotive industry. This conversion not only helps to quantify the engine's performance but also provides a more intuitive understanding of power output when comparing different engines or vehicles.