Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

What constant acceleration will cause a car to increase its velocity from 45 to 60 miles per hour in 10 seconds?

Step-by-Step Solution

Answer

The constant acceleration is 0.67056 m/s².

Step 1: Understand the Problem

To determine the constant acceleration required, we need to know the initial velocity, final velocity, and time interval. We are told the car increases its speed from 45 miles per hour to 60 miles per hour over a period of 10 seconds.

Step 2: Convert Units if Necessary

Since velocity is given in miles per hour and time in seconds, we need to convert the velocities to consistent units, typically meters per second: 1 mile = 1609.34 meters, 1 hour = 3600 seconds.Thus, the initial velocity (\(v_i\)) is 45 mph: \[ v_i = 45 \text{ mph} \times \frac{1609.34 \text{ meters}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 20.1168 \text{ m/s} \]The final velocity (\(v_f\)) is 60 mph: \[ v_f = 60 \text{ mph} \times \frac{1609.34 \text{ meters}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 26.8224 \text{ m/s} \]

Step 3: Use the Formula for Constant Acceleration

The formula to use for constant acceleration is:\[ a = \frac{v_f - v_i}{t} \]where:\\( a \) = acceleration, \\( v_i \) is the initial velocity (20.1168 m/s), \\( v_f \) is the final velocity (26.8224 m/s), and \\( t \) is the time (10 seconds).

Step 4: Calculate the Acceleration

Substitute the known values into the formula:\[ a = \frac{26.8224 \text{ m/s} - 20.1168 \text{ m/s}}{10 \text{ s}} \]Calculate the difference in velocity:\[ a = \frac{6.7056 \text{ m/s}}{10 \text{ s}} \]Finally, calculate the acceleration:\[ a = 0.67056 \text{ m/s}^2 \]

Step 5: Conclude the Problem

The constant acceleration required to increase the car's velocity from 45 to 60 miles per hour in 10 seconds is 0.67056 meters per second squared.

Key Concepts

Velocity Conversion

When dealing with problems that specify velocities in different units, it's important to perform a velocity conversion to ensure consistency. In physics, velocity is typically measured in meters per second (m/s), but in everyday situations, we often see it in miles per hour (mph). Here's an easy way to convert velocity from mph to m/s:

• Start by knowing the conversions: 1 mile = 1609.34 meters and 1 hour = 3600 seconds.
• Multiply the velocity value in mph by 1609.34 to convert miles to meters.
• Then, divide the product by 3600 to convert hours to seconds.

For instance, to convert 45 mph to m/s, the calculation is \( 45 \times \frac{1609.34}{3600} = 20.1168 \, ext{m/s} \). This consistent conversion is crucial for accurately solving problems involving speed and acceleration in physics.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the causes of motion. It's about how objects move and is crucial when discussing velocity, acceleration, and time. Consider a car speeding up or slowing down. Kinematic equations allow us to understand the relationships between the car's initial and final velocities, the time over which the speed change occurs, and the acceleration involved. Kinematics involves equations that link these concepts together. In our original problem, we use one of these key relationships:

• Initial velocity (\( v_i\))
• Final velocity (\( v_f\))
• Time (\( t\))
• Acceleration (\( a\))

Understanding kinematics helps us predict how objects move under constant acceleration, like a car speeding up in this exercise.

Unit Conversion

Unit conversion is a fundamental skill in physics. It’s all about changing a measurement from one unit to another without altering the actual quantity. This skill ensures all measurements are in the appropriate units so they can be accurately used in equations. For example:

• To convert miles to meters, multiply by 1609.34 because there are 1609.34 meters in a mile.
• To convert hours to seconds, multiply by 3600 since one hour equals 3600 seconds.

These conversions are necessary when calculating velocity or acceleration because physics equations often require a specific unit, usually the metric system. Always ensure your units are consistent across all terms within an equation to avoid miscalculations. In practice, changing from miles per hour to meters per second involves both mile-to-meter and hour-to-second conversions, ensuring the speed is correctly understood in a metric context.

Acceleration Formula

The acceleration formula is a critical tool in physics for calculating how quickly the velocity of an object is changing over time. It is defined by the equation:\[ a = \frac{v_f - v_i}{t} \]In this equation:

• \( a \) represents acceleration (in meters per second squared, m/s²).
• \( v_f \) is the final velocity.
• \( v_i \) is the initial velocity.
• \( t \) is the time duration over which the change occurs.

To find out the acceleration, you subtract the initial velocity from the final velocity to find the change in velocity. Then, divide that change by the time period. The outcome tells us how much the velocity increases or decreases each second. In real-world applications, such as speeding up a car, understanding this formula helps us determine the necessary acceleration to reach a desired speed within a set time. This insight is vital in fields like automotive engineering and transportation planning.