3.4: Braking Down the Variables
- Page ID
- 148745
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)INTRODUCTION
In this collaboration, you will be exploring the effects of speed on braking distance of a car. Discuss the following questions in your group:
- Do you stop more quickly when the tires lock and skid or when they do not?
- What is meant by the “grade” of a road?
The formula used in this collaboration uses a subscript for a variable. Subscripts are used in math when there is a variable that we want to represent different values or to give extra information about the variable. For example, if we want to use the variable v to represent the velocity or speed at different times we could use v1 to be the velocity after 1 minute and v2 to be the velocity after 2 minutes. Frequently the subscript zero is used to represent a starting value. So, v0 would be the velocity when the timer is started, or the starting velocity, which is also called the initial velocity.
SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- a variable is a symbol that is used in algebra to represent a quantity that can change.
- many variables can be present in a scenario or experiment, but some can be held fixed in order to analyze the effect that the change in one variable has on another.
By the end of this collaboration, you should be able to
- evaluate an expression.
- informally describe the change in one variable as another variable changes.
PROBLEM SITUATION: CALCULATING THE BRAKING DISTANCE OF A CAR
Experts agree that driving defensively saves lives. Knowing how far it takes your vehicle to come to a complete stop is one aspect of safe driving. For example, when you are going only 45 miles per hour (mph), you are traveling about 66 feet every second. This means that to be a safe driver, you need to drive defensively (i.e., you need to know what is going on ahead of you so that you can react accordingly). In this lesson, you will learn more about what it takes to drive defensively by examining the braking distance of a vehicle. Braking distance is the distance a car travels in the time between when the brake is applied and when it comes to a full stop.
(1) What are some variables that might affect the braking distance of a car? Take a moment to think individually before sharing your ideas in your group. Record some of your answers in the space below.
(2) In building formulas, the factors or conditions are quantified; that is, we find a way to represent the different factors using numbers. The factors or conditions then become the variable in the formula. For example, the steepness of a road, or grade of road, is represented by a percentage of the vertical change divided by the horizontal change. (A 1% grade means you go up 1 foot for every 100 feet of horizontal travel.) For this collaboration, you will examine a variable that affects braking distance: speed. (In Exercise 3.4, you will consider the effects of other variables.)
Discuss with your group how you think speed affects braking distance. Think of some specific questions you might ask. For example, what happens to the braking distance if you double the speed? Would the answer be different for very low speeds or very high speeds? Think about questions individually before sharing in your group. Record your questions in the space below. You will return to these questions later in the collaboration.
Braking Distance: The Formula
The formula for the braking distance of a car is
\[d = \dfrac{V^{2}_{0}}{2g(f + G)}\nonumber \]
V0 = initial velocity of the car (feet per second). That is, the velocity of the car when the brakes were applied. The subscript (symbol written in small type below a variable) zero is used customarily to represent time equaling zero. So, V0 is the velocity when t = 0.
d = braking distance (feet)
G = roadway grade. The grade can be defined as the quantified measure of the change in height of the road (percent written in decimal form). (Note: There are no units for this variable, as explained in the Preparation 3.4.)
f = coefficient of friction between the tires and the roadway (0 < f < 1). (Note: Good tires on good pavement provide a coefficient of friction of about 0.8 to 0.85.)
g = acceleration due to gravity (32.2 ft/sec2)
The “g” in this formula is a constant value; therefore, the formula for braking distance has four variables.
While we normally think about the speed of cars as being measured in miles per hour (mph), in the description for the variables used in the formula above you might have noticed that the velocity (speed at which the car is traveling) is supposed to be measured in feet per second. This is because when we are thinking about the time it takes for a car to stop after hitting the brakes, and how far it travels, we are dealing with small distances and times, and it is more convenient to measure distance and time using feet and seconds rather than miles and hours. For example, when you are going 45 miles per hour (mph), you are traveling about 66 feet every second. You can use dimensional analysis to make this conversion:
\[\dfrac{45\;miles}{hour}\times\dfrac{hour}{60\;minutes}\times\dfrac{minute}{60\;seconds}\times\dfrac{5280\;feet}{mile} = \dfrac{237600\;feet}{3600\;seconds} =\large{66\;feet\;per\;second}\nonumber \]
In this problem situation, you will investigate the relationship between speed when the brakes are applied which is referred to as initial velocity (V0) and braking distance (d). To understand the relationship between these two variables, you will need to hold the other two variables in the formula–friction (f) and roadway grade (G)–fixed. This means that the only variable we are changing is the speed of the car when the brakes are applied. The steepness of the road and the condition of the tires and the road will remain the same. So, we will use the same values for f and G in all of the calculations.
Here is an example of how to use the formula. If a car is traveling at 45 miles per hour and the grade of the road is 1%, and the road and tires are in good condition so coefficient of friction is 0.8, then the formula gives:
\[d = \dfrac{V^{2}_{0}}{2g(f + G)} = \dfrac{66^2}{2(32.2)(0.8+0.01)} = \dfrac{4356}{52.164} = \large{83.5\;feet}\nonumber \]
(3) Let f = 0.85 (good tires on good roads) and G = 0.06 (slight uphill road grade). We also have g = acceleration due to gravity is 32.2 ft/sec2. Use these values in the braking distance formula to calculate the braking distance for a car traveling at 75 mph. Take a moment to think on your own first before sharing in your group. Write your answer below rounded to one decimal place. Note: remember to convert miles per hour in feet per second!
(4) Let f = 0.8 (good tires on good roads) and G = 0.05 (slight uphill road grade). Use these values for f and G to simplify the braking distance formula. Take a moment to think on your own first before sharing in your group. Note: You also know the value for g, because the acceleration due to gravity is constant: g = acceleration due to gravity.
(5) How can you verify your predictions about the relationship between velocity and braking distance?
(6) Now you will explore the question(s) developed by the class in Question 2.
(a) Record the question(s) here:
(b) Create a strategy for exploring the question(s) with your group. Record your strategies here:
(c) Use your strategies to answer the question(s). Write a complete statement about your results here:
The three Skills You Need In Any Field
You have now used several different formulas in this course. In Lesson 3.3, you used common geometric formulas for area and volume (you probably have seen these formulas before). In this lesson, you used a formula that was more complex and probably less familiar to you. Almost every field has specialized formulas, but they all depend on three basic skills:
- Understanding and knowing how to use variables, including the use of subscripts.
- Understanding and knowing how to use the order of operations.
- Understanding and knowing how to use units, including dimensional analysis.
With these three skills, you will be able to use formulas in any field.
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.