How far will it travel along the horizontal surface before it stops?

Question

A 3.2 kg crates slides along a 6.0 m horizontal surface with speed of 5.0m/s. It then moves down along an incline with a lenth 6.0m and a height from the ground of 4.5 mand then along a second, lower horizontal surface. The second horizontal surface has a frictional force of 3.0 N. How far will it travel along the second horizontal surface before it stops?

Answer 1

I will explain the steps, and you do the math.  I assume the first horizontal slide and the slide down the incline are without friction.

 

Initial speed at top of incline: v₁ = 5.0 m/s

 

Get speed v₂ at bottom of incline using conservation of energy.

 

v₁²/2 + gh = v₂²/2

 

v₂ = √[2(v₁²/2 + gh)]

 

You know v₁, g = 9.8 m/s², and h = 4.5 m.  Plug in the numbers and get v₂.

 

Next, find the acceleration a along the second horizontal slide, with friction.

 

a = F/m = (-3.0 N)/(3.2 kg) = ? m/s²

 

The acceleration is negative because the force of friction is opposite the direction of motion.  Plug in the numbers and get a.

 

Next, get the stopping distance d using the kinematic relation

 

v₃² - v₂² = 2ad

 

You have v₂ and a, and you know v₃ = 0 (comes to a full stop).

 

-v₂² = 2ad

 

d = -v₂²/(2a)

 

Since a is negative, the distance d will come out positive.  You have all you need to finish from here.