# Find the Stopping Distance with Calculus

## 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!

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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.