# A box with an initial speed of #1 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #2/3 # and an incline of #(3 pi )/8 #. How far along the ramp will the box go?

The kinetic friction force and the parallel component of the gravitational force oppose each other as the box moves forward; that is, they are both negatively-signed with respect to forward motion.

So:

Consequently,

and

Consequently:

Assuming the ramp is straight, this indicates that the box will logically slow down to zero velocity. This gives us an estimate of the average acceleration:

Consequently,

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To find the distance along the ramp (s), use the equation:

[ s = \frac{v_0^2}{2g} \left(\frac{1}{\mu_k} - \tan(\theta)\right) ]

where: ( v_0 = 1 \ \text{m/s} ) (initial speed), ( g = 9.8 \ \text{m/s}^2 ) (acceleration due to gravity), ( \mu_k = \frac{2}{3} ) (kinetic friction coefficient), ( \theta = \frac{3\pi}{8} ) (inclination of the ramp).

Plug in the values to find the distance along the ramp.

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