The Stopping Distance Table

Oct 13, 2025 by pynkocean.com 37 views

Introduction

The stopping distance of a vehicle is a critical factor in road safety, as it determines the minimum distance required for a vehicle to come to a complete stop after a driver has applied the brakes. In this article, we will analyze the stopping distance problem using mathematical techniques, with a focus on the data provided in the table below.

The Stopping Distance Table

Speed (v)	Stopping Distance (d)
20	38
30	86
40	153

Problem Statement

The problem statement is to find a mathematical model that describes the relationship between the speed of a vehicle and its stopping distance. In other words, we want to find a function that takes the speed of a vehicle as input and returns the corresponding stopping distance.

Linear Regression Analysis

One possible approach to solving this problem is to use linear regression analysis. Linear regression is a statistical technique that models the relationship between a dependent variable (in this case, the stopping distance) and one or more independent variables (in this case, the speed of the vehicle).

To perform a linear regression analysis, we need to calculate the mean and standard deviation of the stopping distance and speed data. We also need to calculate the correlation coefficient between the two variables.

Calculating the Mean and Standard Deviation

The mean of the stopping distance data is calculated as follows:

\bar{d} = \frac{38 + 86 + 153}{3} = 92.33

The standard deviation of the stopping distance data is calculated as follows:

s_d = \sqrt{\frac{(38-92.33)^2 + (86-92.33)^2 + (153-92.33)^2}{3-1}} = 43.15

The mean of the speed data is calculated as follows:

\bar{v} = \frac{20 + 30 + 40}{3} = 30

The standard deviation of the speed data is calculated as follows:

s_v = \sqrt{\frac{(20-30)^2 + (30-30)^2 + (40-30)^2}{3-1}} = 10

Calculating the Correlation Coefficient

The correlation coefficient between the stopping distance and speed data is calculated as follows:

r = \frac{\sum_{i=1}^n (d_i - \bar{d})(v_i - \bar{v})}{\sqrt{\sum_{i=1}^n (d_i - \bar{d})^2 \sum_{i=1}^n (v_i - \bar{v})^2}}

where $d_i$ and $v_i$ are the individual data points, and $n$ is the number of data points.

Plugging in the values, we get:

r = \frac{(38-92.33)(20-30) + (86-92.33)(30-30) + (153-92.33)(40-30)}{\sqrt{(38-92.33)^2 + (86-92.33)^2 + (153-92.33)^2} \sqrt{(20-30)^2 + (30-30)^2 + (40-30)^2}}

r = \frac{-54.33 + 0 + 60.67}{43.15 \times 10}

r = \frac{6.34}{431.5}

r = 0.0146

Linear Regression Model

The linear regression model is given by the equation:

d = \beta_0 + \beta_1 v + \epsilon

where $d$ is the stopping distance, $v$ is the speed, $\beta_0$ and $\beta_1$ are the regression coefficients, and $\epsilon$ is the error term.

To estimate the regression coefficients, we can use the ordinary least squares (OLS) method. The OLS method minimizes the sum of the squared errors between the observed and predicted values.

The estimated regression coefficients are:

\hat{\beta_0} = 92.33 - 0.0146 \times 30 = 92.33 - 0.4398 = 91.89

\hat{\beta_1} = 0.0146

The linear regression model is:

d = 91.89 + 0.0146 v

Interpretation of Results

The linear regression model suggests that the stopping distance is positively related to the speed of the vehicle. This means that as the speed of the vehicle increases, the stopping distance also increases.

The regression coefficient $\hat{\beta_1}$ represents the change in stopping distance for a one-unit change in speed. In this case, the regression coefficient is 0.0146, which means that for every 1 km/h increase in speed, the stopping distance increases by approximately 0.0146 meters.

Conclusion

In this article, we analyzed the stopping distance problem using linear regression analysis. We calculated the mean and standard deviation of the stopping distance and speed data, and estimated the regression coefficients using the OLS method. The linear regression model suggests that the stopping distance is positively related to the speed of the vehicle.

Limitations of the Study

One limitation of this study is that it is based on a small sample size of only three data points. A larger sample size would provide more reliable estimates of the regression coefficients and would allow for a more accurate model of the relationship between the stopping distance and speed.

Another limitation of this study is that it assumes a linear relationship between the stopping distance and speed. In reality, the relationship may be non-linear, and a more complex model may be needed to accurately capture the relationship.

Future Research Directions

Future research directions could include:

Collecting a larger sample size of data points to improve the accuracy of the regression coefficients
Using a more complex model, such as a quadratic or cubic model, to capture the non-linear relationship between the stopping distance and speed
Incorporating additional variables, such as the type of vehicle or the road surface, into the model to improve its accuracy

References

[1] National Highway Traffic Safety Administration. (2020). Traffic Safety Facts: 2019 Data.
[2] Federal Highway Administration. (2020). Highway Statistics 2019.
[3] American Automobile Association. (2020). Traffic Safety Report: 2019 Data.
Q&A: Stopping Distance Problem =====================================

Q: What is the stopping distance problem?

A: The stopping distance problem is a mathematical problem that involves finding a relationship between the speed of a vehicle and its stopping distance. The stopping distance is the minimum distance required for a vehicle to come to a complete stop after a driver has applied the brakes.

Q: Why is the stopping distance problem important?

A: The stopping distance problem is important because it can help us understand how to design safer roads and vehicles. By knowing the relationship between speed and stopping distance, we can design roads and vehicles that are more resistant to accidents.

Q: What are some common factors that affect the stopping distance?

A: Some common factors that affect the stopping distance include:

Speed: The faster a vehicle is traveling, the longer it will take to stop.
Road surface: The type of road surface can affect the stopping distance. For example, a vehicle may stop more quickly on a dry road than on a wet road.
Vehicle type: The type of vehicle can also affect the stopping distance. For example, a vehicle with a heavier weight may stop more slowly than a lighter vehicle.
Driver behavior: The behavior of the driver can also affect the stopping distance. For example, a driver who is distracted or not paying attention may not be able to stop as quickly as a driver who is fully attentive.

Q: How can we use linear regression to model the stopping distance problem?

A: We can use linear regression to model the stopping distance problem by using the following equation:

d = β0 + β1v + ε

where d is the stopping distance, v is the speed, β0 and β1 are the regression coefficients, and ε is the error term.

Q: What are the limitations of using linear regression to model the stopping distance problem?

A: One limitation of using linear regression to model the stopping distance problem is that it assumes a linear relationship between the stopping distance and speed. In reality, the relationship may be non-linear, and a more complex model may be needed to accurately capture the relationship.

Another limitation is that the model may not account for other factors that affect the stopping distance, such as road surface and vehicle type.

Q: What are some potential applications of the stopping distance problem?

A: Some potential applications of the stopping distance problem include:

Designing safer roads and vehicles
Developing more accurate traffic models
Improving driver training programs
Reducing the number of accidents on the road

Q: How can we improve the accuracy of the stopping distance model?

A: We can improve the accuracy of the stopping distance model by:

Collecting more data points to improve the sample size
Using a more complex model, such as a quadratic or cubic model, to capture the non-linear relationship between the stopping distance and speed
Incorporating additional variables, such as road surface and vehicle type, into the model to improve its accuracy

Q: What are some potential future research directions for the stopping distance problem?

A: Some potential future research directions for the stopping distance problem include:

Developing more accurate traffic models that take into account the stopping distance
Improving driver training programs to reduce the number of accidents on the road
Designing safer roads and vehicles that are more resistant to accidents
Developing more complex models that can capture the non-linear relationship between the stopping distance and speed.

Q: How can we apply the stopping distance problem to real-world scenarios?

A: We can apply the stopping distance problem to real-world scenarios by:

Using the model to design safer roads and vehicles
Developing more accurate traffic models that take into account the stopping distance
Improving driver training programs to reduce the number of accidents on the road
Reducing the number of accidents on the road by designing safer roads and vehicles.

Q: What are some potential challenges and limitations of applying the stopping distance problem to real-world scenarios?

A: Some potential challenges and limitations of applying the stopping distance problem to real-world scenarios include:

Collecting accurate data on the stopping distance and speed
Developing a model that can accurately capture the non-linear relationship between the stopping distance and speed
Incorporating additional variables, such as road surface and vehicle type, into the model to improve its accuracy
Addressing the limitations of the model, such as its assumption of a linear relationship between the stopping distance and speed.