Answer to Question #243248 in Physics for megs

Question

A traveler drops her 46 kg suitcase onto a surface that is angled at 20° above the horizontal. The length of the angled surface is 7.9 m. The coefficient of kinetic friction of the surface is 0.31. How long will it take for the suitcase to slide down the ramp if its initial velocity after it is dropped is zero?

Answer 1

Apply Newton's second law:

$$x:ma=mg\sin\theta-f,\\ y: 0=N-mg\cos\theta.\\\space\\ N=mg\cos\theta,\\ f=\mu N=\mu mg\cos\theta. \\\space\\ a=\frac{mg\sin\theta-\mu mg\cos\theta}{m}=g(\sin\theta-\mu\cos\theta).$$

The time it will take to reach the bottom is

$$t=\sqrt{\frac{2d}{a}}=\sqrt{\frac{2d}{g(\sin\theta-\mu\cos\theta)}}=5.6\text{ s}.$$