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?

Answer 1
I would take the sum of the parallel and perpendicular forces. The perpendicular forces (#vecF_N#, #vecF_g#) cancel out.

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:

#sum vecF_(||) = -vecF_k - vecF_(g,||)#
#= -mu_kvecF_N - mvecgsintheta = mveca_(||)#
where we have put the signs in the equation, #mu_k = 2/3#, and #vecg > 0#.
#sum vecF_(_|_) = vecF_N - vecF_(g,_|_) = vecF_N - mvecgcostheta = 0#

Consequently,

#vecF_N = mvecgcostheta#

and

#-mu_kcancel(m)vecgcostheta - cancel(m)vecgsintheta = cancel(m)veca_(||)#

Consequently:

#veca_(||) = -(mu_k vecg cos theta + vecg sin theta)#
#= -(2/3 cdot "9.81 m/s"^2 cdot cos((3pi)/8) + "9.81 m/s"^2 cdot sin((3pi)/8))#
#= -"11.57 m/s"^2#

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:

#veca_(||) -= (Deltavecv_(||))/(Deltax)#
Letting the bottom of the ramp be the initial position #x_i = "0 m"# and the final velocity being #"0 m/s"#,
#veca_(||) = (0 - vecv_i)/(x_f - 0)#

Consequently,

#color(blue)(x_f) = -(vecv_i)/(veca_(||)) = -("1 m/s")/(-"11.57 m/s"^2)#
#=# #color(blue)"0.086 m"# up the ramp.
That's about #3.4# inches.
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Answer 2

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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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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