Question

A box with an initial speed of `8 m/s` is moving up a ramp. The ramp has a kinetic friction coefficient of `2/5` and an incline of `pi /4`. How far along the ramp will the box go?

Answer 1

The distance is `=3.30m`

Explanation:

Resolving in the direction up and parallel to the plane as positive `↗^+`

The coefficient of kinetic friction is `mu_k=F_r/N`

Then the net force on the object is

`F=-F_r-Wsintheta`

`=-F_r-mgsintheta`

`=-mu_kN-mgsintheta`

`=mmu_kgcostheta-mgsintheta`

According to Newton's Second Law of Motion

`F=m*a`

Where `a` is the acceleration of the box

So

`ma=-mu_kgcostheta-mgsintheta`

`a=-g(mu_kcostheta+sintheta)`

The coefficient of kinetic friction is `mu_k=2/5`

The acceleration due to gravity is `g=9.8ms^-2`

The incline of the ramp is `theta=1/4pi`

The acceleration is `a=-9.8*(2/5cos(1/4pi)+sin(1/4pi))`

`=-9.7ms^-2`

The negative sign indicates a deceleration

Apply the equation of motion

`v^2=u^2+2as`

The initial velocity is `u=8ms^-1`

The final velocity is `v=0`

The acceleration is `a=-9.7ms^-2`

The distance is `s=(v^2-u^2)/(2a)`

`=(0-64)/(-2*9.7)`

`=3.30m`