With a constant braking force, the stopping distance of a car is proportional to
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The stopping distance of an automobile, on dry, level pavement, traveling at a speed $v$ (in kilometers per hour) is the distance $R$ (in meters) the car travels during the reaction time of the driver plus the distance $B$ (in meters) the car travels after the brakes are applied (see figure). The table shows the results of an experiment. (IMAGE CANNOT COPY) (TABLE CANNOT COPY) (a) Use the regression capabilities of a graphing utility to find a linear model for reaction time distance $R$. (b) Use the regression capabilities of a graphing utility to find a quadratic model for braking time distance $B$. (c) Determine the polynomial giving the total stopping distance $T$. (d) Use a graphing utility to graph the functions $R, B,$ and $T$ in the same viewing window. (e) Find the derivative of $T$ and the rates of change of the total stopping distance for $v=40, v=80,$ and $v=100$. (f) Use the results of this exercise to draw conclusions about the total stopping distance as speed increases.
Differentiation
Basic Differentiation Rules and Rates of Change
Braking Distance The table below gives the results of an online calculator showing how far (in feet) a vehicle will travel while braking to a complete stop, given the initial velocity of the automobile. Braking Distance $$ \begin{array}{c|c} \text { MPH } & \text { Distance (feet) } \\ \hline 10 & 27 \\ \hline 20 & 63 \\ \hline 30 & 109 \\ \hline 40 & 164 \\ \hline 50 & 229 \\ \hline 60 & 304 \\ \hline 70 & 388 \\ \hline 80 & 481 \\ \hline 90 & 584 \\ \hline \end{array} $$ a. Find a quadratic model for the stopping distance. b. What other factors besides the initial speed would affect the stopping distance?
Ingredients of Change: Functions and Limits
Quadratic Functions and Models
The accompanying table shows the distance a car travels during the time the driver is reacting before applying the brakes, and the distance the car travels after the brakes are applied. The distances (in feet) depend on the speed of the car (in miles per hour). Test the reasonableness of the following proportionality assumptions and estimate the constants of proportionality. a. reaction distance is proportional to speed. b. braking distance is proportional to the square of the speed.
Preliminaries
Identifying Functions; Mathematical Models
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
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