# Find the Stopping Distance with Calculus

• kchurchi
In summary, to solve the problem of finding the minimal length of road needed for a truck to stop without having a 290-N crate slide, one can use the coefficient of static friction and the coefficient of kinetic friction between the crate and the truck. By knowing the maximum amount of force the crate can take before slipping, the maximum acceleration of the truck can be determined, and from there, the length of road needed can be calculated using one of the SUVAT equations.
kchurchi

## Homework Statement

A 290-N crate rides without constraints on the horizontal floor of a truck. The coefficient of static friction and the coefficient of kinetic friction between the crate and the truck are 0.32 and 0.16, respectively. If the truck is initially traveling at 18 m/s, what minimal length of road is needed for it to stop without having the crate slide?

Ʃ F(net) = m*a

## The Attempt at a Solution

I drew free body diagrams of the crate, of the truck, and of the crate plus the truck. I plan on using the x-acceleration to solve for a velocity expression. When I set that velocity expression equal to zero I can find the time it takes to stop. I can use this time to find the distance. HOWEVER I can't figure out my friction force. The only force I got acting in the x-direction is friction from the truck on the crate or from the crate on the truck. I know it isn't kinetic friction because it doesn't slip, but I don't know what to do? Help!

Knowing the maximum amount of force the crate can take before slipping should give you all you need to know for this problem.

In other words, since you know the coefficient of static friction between the crate and the truck, then you should be able to find out the maximum acceleration that the truck can reach before the force between the bed of the truck and the crate overcomes the static friction acting on the crate.

Wouldn't that give me the maximum stopping distance? Because I tried doing that and didn't get the right answer.

You do not need time to solve this problem.
Use one of the SUVAT equations.
What is the maximum acceleration if there is no slippage?

I would use the SUVAT equations but I need to show I derived them, I can't just pick one unfortunately.

Find all available data or derived data and find one that suits.
Anyway there are only 3 of them.

## 1. What is the stopping distance?

The stopping distance is the total distance traveled by a moving object from the moment the brakes are applied until the object completely stops.

## 2. How is calculus used to find the stopping distance?

Calculus is used to find the stopping distance by integrating the deceleration function over time. This results in a position function, which can then be solved for the distance traveled when the object comes to a stop.

## 3. What factors affect the stopping distance?

The stopping distance is affected by several factors, including the initial speed of the object, the coefficient of friction between the object and the surface it is traveling on, and the reaction time of the person applying the brakes.

## 4. Why is it important to calculate the stopping distance?

Calculating the stopping distance is important for ensuring the safety of drivers and passengers. It can also help engineers design safer vehicles and roadways by understanding the factors that affect stopping distance.

## 5. How can calculus be applied to real-world situations involving stopping distance?

Calculus is frequently used in real-world situations involving stopping distance, such as in traffic engineering to design intersections and in vehicle design to improve braking systems. It can also be used to analyze the effectiveness of different road surfaces and to determine appropriate speed limits.

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