The stopping distance of an automobile, on dry, level...

The stopping distance of an automobile, on dry, level pavement, traveling at a speed $v$ (kilometers per hour) is the distance $R$ (meters) the car travels during the reaction time of the driver plus the distance $B$ (meters) the car travels after the brakes are applied (see figure). The table shows the results of an experiment. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Speed, } \boldsymbol{v} & 20 & 40 & 60 & 80 & 100 \\\hline \begin{array}{l}\text { Reaction Time } \\\text { Distance, } \boldsymbol{R}\end{array} & 8.3 & 16.7 & 25.0 & 33.3 & 41.7 \\\hline \begin{array}{l}\text { Braking Time } \\\text { Distance, } \boldsymbol{B}\end{array} & 2.3 & 9.0 & 20.2 & 35.8 & 55.9 \\\hline\end{array}$$ (a) Use the regression capabilities of a graphing utility to find a linear model for reaction time distance. (b) Use the regression capabilities of a graphing utility to find a quadratic model for braking distance. (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.

The stopping distance of an automobile, on dry, level pavement, traveling at a speed $v$ (kilometers per hour) is the distance $R$ (meters) the car travels during the reaction time of the driver plus the distance $B$ (meters) the car travels after the brakes are applied (see figure). The table shows the results of an experiment.
$$\begin{array}{|l|c|c|c|c|c|}\hline \text { Speed, } \boldsymbol{v} & 20 & 40 & 60 & 80 & 100 \\\hline \begin{array}{l}\text { Reaction Time } \\\text { Distance, } \boldsymbol{R}\end{array} & 8.3 & 16.7 & 25.0 & 33.3 & 41.7 \\\hline \begin{array}{l}\text { Braking Time } \\\text { Distance, } \boldsymbol{B}\end{array} & 2.3 & 9.0 & 20.2 & 35.8 & 55.9 \\\hline\end{array}$$
(a) Use the regression capabilities of a graphing utility to find a linear model for reaction time distance.
(b) Use the regression capabilities of a graphing utility to find a quadratic model for braking distance.
(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.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes with respect to changes in its input variable. In the context of stopping distance, differentiating the total distance function with respect to speed provides insight into how sensitive the stopping distance is to small changes in speed.

Interpretation of Results

Interpreting the results involves understanding the practical implications of the mathematical models. As speed increases, both the reaction distance and especially the braking distance increase, leading to a non-linear (often accelerating) growth in total stopping distance. This has significant implications for vehicle safety guidelines and accident prevention strategies.

Polynomial Modeling

Polynomial modeling refers to combining simpler models (like the linear and quadratic models) into a single, more complex model that describes the total stopping distance as a function of speed. This involves adding the components modeled separately to obtain a composite function that reflects the overall behavior.

Rate of Change Analysis

Rate of change analysis involves evaluating the derivative of the total stopping distance function at specific speeds. This analysis helps in understanding how quickly the stopping distance increases as speed changes, offering a numerical measure of vehicle responsiveness and safety profile under different conditions.

Linear Regression

Linear regression involves fitting a straight line to a set of data points. For reaction time distance, the assumption is that this distance increases at a constant rate with speed, making a linear model appropriate to capture the direct relationship between speed and distance covered during the reaction period.

Regression Analysis

Regression analysis is a statistical process for estimating the relationships among variables. In this context, it is used to determine the best-fit model that describes how stopping distances change with speed, allowing for predictions and analysis of performance under various conditions.

Quadratic Regression

Quadratic regression fits a quadratic equation (a parabola) to the data points. This method is used when the relationship between the braking distance and speed implies that the distance increases at an increasing rate, which is modeled effectively by a quadratic function that represents the impact of squared terms.

Transcript

00:01 And this problem, they give us a chart here that i have reproduced.

00:09 And they give us a bunch of speeds.

00:11 So in increments of 20, they tell us how the kilometers per hour, the distance of 20, 40, 60, 80, and 100 kilometers per hour.

00:21 And then they tell us this first row here is the distance that we go before we would hit hit the break.

00:31 So basically it's our reaction time distance.

00:33 So we see something we want to stop.

00:37 And so there's some reaction time before we start the actual break, actually break.

00:42 And then once we start to actually break, we then slow down.

00:46 So what this shows is if we, if we plot this, you can see this is very linear.

00:53 And that's just because our reaction time is pretty much the same, no matter what speed we're going.

00:58 So the distance we travel, increases linearly with the speed.

01:07 So if we plot that, we get very close to a line.

01:12 And if we fit that line, as they said, s is due, we get 0 .417 times the velocity minus 0 .02.

01:23 So this is very close to 0.

01:25 We would expect that, you know, if v was zero, then this should probably be 0.

01:30 But it was this little small offset because of the, you know, just kind of the noise and the data.

01:37 You might set that to zero for just based on, you know, that the reaction time distance should be zero when the speed is zero.

01:48 And so then the braking, so this is the distance that we travel after we start breaking.

01:56 And if we plot this, it looks very quadratic.

01:59 And that's basic because you're dissipating energy and the energy increase.

02:03 With the velocity squared.

02:06 So to get rid of all the energy, you would expect to have a quadratic function so that this is going to grow quadratically with v squared.

02:14 And indeed it does.

02:16 And so we plot this, and we make a quadratic fit, quadratic model, and we get 0 .00557 v squared plus 0 .00143v plus 0 .043v and again it makes it.

02:34 For physical reasons, you might say this should be zero, because if v is zero, then we would expect this to be zero.

02:44 But again, we can, whether you keep this or not, it's a very minor offset when you're talking about these other terms and as far as their magnitude goes.

02:58 So the total distance then is just the sum of these two.

03:05 So we just add them up.

03:07 We get this quadratic term.

03:09 This is pretty much dominates the linear term compared to that.

03:15 And then we get these, you know, this offset here.

03:18 And again, we could maybe say that was zero for, just for physical reasons that if you're not traveling any, if you're not moving, then your stopping distance should be zero.

03:29 But again, this is very small compared to these when you plug in, you know, these respective values here...

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