Help with Kinetic Energy and Time

AI Thread Summary
To determine the shortest time for a pickup truck to accelerate to 32.0 m/s without the toolbox sliding out, first calculate the maximum static friction force using the coefficient of static friction (0.700) and the weight of the box. Apply Newton's second law to find the maximum acceleration that static friction can provide. Once the maximum acceleration is established, use the kinematic equation to solve for the time needed to reach the target speed of 32.0 m/s. A force diagram can aid in visualizing the forces acting on the box during acceleration. Understanding these concepts is crucial for solving the problem effectively.
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A pickup truck is carrying a toolbox, but the rear gate of the truck is missing, so the box will slide out if it is set moving. The coefficients of kinetic and static friction between the box and the bed of the truck are 0.305 and 0.700, respectively.

Starting from rest, what is the shortest time this truck could accelerate uniformly to 32.0 ( 71.6) without causing the box to slide. (Hint: First use Newton’s second law to find the maximum acceleration that static friction can give the box, and then solve for the time required to reach 32.0 .)

I am not sure what formulas to use or how to solve the problem, please help. Thanks.
 
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Draw a force diagram of the box while the truck is accelerating, but when the box isn't sliding. Then just balance forces.
 

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