A coffee cup on the horizontal dashboard of a car slides forward when the driver decelerates from 45 kmh to rest in 3.5 s or less, but not if she decelerates in a longer time. What is the coefficient of static friction between the cup and the dash? Assume the road and the dashboard is level (horizontal).

μs = 0.36

Explanation:

Assuming no other forces acting on the cup while the car is decelerating, the friction force is responsible for any horizontal movement of the cup.

If the cup is on the verge of starting to slide, the friction force can be expressed as follows:

Ff = -μs*N = -μs*m*g

This force produces a deceleration from 45 Kmh to rest, in 3.5 s or  more.

Converting 45 kmh to m/s, we have:

We can find the acceleration, just applying the definition, with vf =0, as follows:

According to Newton's 2nd law, we can write the following expression:

F = m*a = -μs*m*g

Simplifying common terms, we can solve for μs, as follows:

μs =