Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

If the brakes of a car, when fully applied, produce a constant deceleration of 11 feet per second per second, what is the shortest distance in which the car can be braked to a halt from a speed of 60 miles per hour?

Step-by-Step Solution

Answer

The car can be braked to a halt in 352 feet.

Step 1: Convert Speed to Consistent Units

First, convert the speed from miles per hour to feet per second. Since 1 mile = 5280 feet and 1 hour = 3600 seconds, the conversion factor is \( \frac{5280}{3600} \). Therefore, to convert 60 miles per hour to feet per second: \( 60 \times \frac{5280}{3600} = 88 \) feet per second.

Step 2: Use the Deceleration Formula

The formula for deceleration to a complete stop is \( v^2 = u^2 + 2as \), where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration (negative in this context for deceleration), and \( s \) is the distance. Since the final velocity \( v = 0 \) (the car stops), and the initial velocity \( u = 88 \) feet per second, and \( a = -11 \) feet per second squared, the equation simplifies to \( 0 = 88^2 + 2(-11)s \).

Step 3: Solve for the Distance

Rearrange the equation to solve for \( s \): \( 0 = 88^2 - 22s \). This becomes \( 22s = 88^2 \). Calculate \( 88^2 = 7744 \), then solve for \( s \): \( s = \frac{7744}{22} \). Division gives \( s = 352 \) feet.

Key Concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects. It describes how objects move with respect to time, without considering the forces that cause these movements. By understanding kinematics, you can analyze different types of motion, such as linear and rotational. In the context of the car braking problem, we deal with linear motion.
The car is initially moving and must decelerate to a stop. Hence, kinematics principles apply, specifically the relationship between velocity, acceleration (or deceleration, in this case), and distance. These concepts are crucial in calculating how far the car travels during its deceleration phase.

Unit Conversion

In physics problems, making sure all units are consistent is crucial. Without converting units to a common base, the equations used can yield incorrect results. In this exercise, we deal with converting speed from miles per hour to feet per second.
Here's the step-by-step conversion process:

• 1 mile = 5280 feet and 1 hour = 3600 seconds.
• For speed in miles per hour: Multiply by 5280/3600 to convert to feet per second.

This step ensures the speed aligns with the units of deceleration, which are already in feet per second squared. This precise conversion aids in accurate calculations in subsequent steps of the problem.

Physics Equations

Physics equations like the deceleration formula are tools to describe and predict motion. The equation used here is derived from one of the basic kinematic equations:

• The formula is: \[ v^2 = u^2 + 2as \] where \( v \) is final velocity, \( u \) is initial velocity, \( a \) is acceleration, and \( s \) is distance covered.
• In the car deceleration problem, final velocity \( v = 0 \) because the car stops.

This equation is remodeled to find the distance using known values of initial velocity, deceleration, and the fact that the car comes to a complete halt. By solving this equation, we can determine how far the car travels before stopping.

Distance Calculation

Calculating distance involves rearranging and solving the kinematics equation specific to deceleration. Using the given deceleration formula:

• Insert the known values into the equation: \[ 0 = 88^2 + 2(-11)s \]
• Solve for \( s \) by isolating it: \( 0 = 7744 - 22s \), then \( 22s = 7744 \).

The result of this calculation shows the shortest stopping distance. Solve this by dividing: \( s = \frac{7744}{22} \), which equals 352 feet. This process illustrates the power of kinematic equations in predicting real-world outcomes, offering precise results.