A box with an initial speed of $8 \frac{m}{s}$ $8 m/s$ is moving up a ramp. The ramp has a kinetic friction coefficient of $\frac{4}{3}$ $4/3$ and an incline of $\frac{π}{6}$ $( pi )/6$ . How far along the ramp will the box go?

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A box with an initial speed of $8 \frac{m}{s}$ $8 m/s$ is moving up a ramp. The ramp has a kinetic friction coefficient of $\frac{4}{3}$ $4/3$ and an incline of $\frac{π}{6}$ $( pi )/6$ . How far along the ramp will the box go?

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Answer 1 · 2017-07-11T08:01:00

The distance is $= 1.97 m$ $=1.97m$

Explanation:

Taking the direction up and parallel to the plane as positive $↗^{+}$ $↗^+$

The coefficient of kinetic friction is $μ_{k} = \frac{F_{r}}{N}$ $mu_k=F_r/N$

Then the net force on the object is

$F = - F_{r} - W sin θ$ $F=-F_r-Wsintheta$

$= - F_{r} - m g sin θ$ $=-F_r-mgsintheta$

$= - μ_{k} N - m g sin θ$ $=-mu_kN-mgsintheta$

$= m μ_{k} g cos θ - m g sin θ$ $=mmu_kgcostheta-mgsintheta$

According to Newton's Second Law

$F = m \cdot a$ $F=m*a$

Where $a$ $a$ is the acceleration

$m a = - μ_{k} g cos θ - m g sin θ$ $ma=-mu_kgcostheta-mgsintheta$

$a = - g (μ_{k} cos θ + sin θ)$ $a=-g(mu_kcostheta+sintheta)$

The coefficient of kinetic friction is $μ_{k} = \frac{4}{3}$ $mu_k=4/3$

The incline of the ramp is $θ = \frac{1}{6} π$ $theta=1/6pi$

$a = - 9.8 \cdot (\frac{4}{3} cos (\frac{1}{6} π) + sin (\frac{1}{6} π))$ $a=-9.8*(4/3cos(1/6pi)+sin(1/6pi))$

$= - 16.22 m s^{- 2}$ $=-16.22ms^-2$

The negative sign indicates a deceleration

We apply the equation of motion

$v^{2} = u^{2} + 2 a s$ $v^2=u^2+2as$

$u = 8 m s^{- 1}$ $u=8ms^-1$

$v = 0$ $v=0$

$a = - 16.22 m s^{- 2}$ $a=-16.22ms^-2$

$s = \frac{v^{2} - u^{2}}{2 a}$ $s=(v^2-u^2)/(2a)$

$= \frac{0 - 64}{- 2 \cdot 16.02}$ $=(0-64)/(-2*16.02)$

$= 1.97 m$ $=1.97m$

NOTE : The coefficient of friction $μ_{k} > 1$ $mu_k>1$ means that the frictional force is greater than the normal force.

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A box with an initial speed of $8 \frac{m}{s}$ $8 m/s$ is moving up a ramp. The ramp has a kinetic friction coefficient of $\frac{4}{3}$ $4/3$ and an incline of $\frac{π}{6}$ $( pi )/6$ . How far along the ramp will the box go?

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Explanation:

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A box with an initial speed of 8ms8 m/s is moving up a ramp. The ramp has a kinetic friction coefficient of 434/3 and an incline of π6( pi )/6 . How far along the ramp will the box go?

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A box with an initial speed of $8 \frac{m}{s}$ $8 m/s$ is moving up a ramp. The ramp has a kinetic friction coefficient of $\frac{4}{3}$ $4/3$ and an incline of $\frac{π}{6}$ $( pi )/6$ . How far along the ramp will the box go?