Velocity on an Incline?

Question

A 200 N box slides down an incline that is 5.0 m long and drops 1.0 m. The incline has a coefficient of kinetic friction of 0.1. What is the velocity at the bottom of the incline?

Answer 1

Let:

m be the mass of the box,

g be the acceleration due to gravity,

a be the acceleration of the box,

F be the friction force up the incline,

R be the normal reaction of the incline on the box,

u be the coefficient of kinetic friction,

t be the angle of the slope from the horizontal,

v be the velocity of the box at the bottom of the slope.

Resolving parallel and perpendicular to the slope:

mg sin(t) - uR = ma

mg cos(t) = R

Eliminating R:

mg sin(t) - umg cos(t) = ma

g[ sin(t) - u cos(t) ] = a

v^2 = 2as

= 2gs[ sin(t) - u cos(t) ]

sin(t) = 0.2, cos(t) = sqrt(1 - 0.2^2) = 0.9798.

v = sqrt[ 2 * 9.81 * 5.0(0.2 - 0.1 * 0.9798 ]

= 3.16 m/s.