A box with an initial speed of $3 \frac{m}{s}$ $3 m/s$ is moving up a ramp. The ramp has a kinetic friction coefficient of $\frac{8}{7}$ $8/7$ and an incline of $\frac{π}{4}$ $( pi )/4$ . How far along the ramp will the box go?

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A box with an initial speed of $3 \frac{m}{s}$ $3 m/s$ is moving up a ramp. The ramp has a kinetic friction coefficient of $\frac{8}{7}$ $8/7$ and an incline of $\frac{π}{4}$ $( pi )/4$ . How far along the ramp will the box go?

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Answer 1 · 2017-06-06T06:43:23

The distance is $= 0.30 m$ $=0.30m$

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

So

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

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

The coefficient of kinetic friction is $μ_{k} = \frac{8}{7}$ $mu_k=8/7$

The incline of the ramp is $θ = \frac{1}{4} π$ $theta=1/4pi$

$a = - 9.8 \cdot (\frac{8}{7} cos (\frac{1}{4} π) + sin (\frac{1}{4} π))$ $a=-9.8*(8/7cos(1/4pi)+sin(1/4pi))$

$= - 14.85 m s^{- 2}$ $=-14.85ms^-2$

The negative sign indicates a deceleration

We apply the equation of motion

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

$u = 3 m s^{- 1}$ $u=3ms^-1$

$v = 0$ $v=0$

$a = - 11.16 m s^{- 2}$ $a=-11.16ms^-2$

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

$= \frac{0 - 9}{- 2 \cdot 14.85}$ $=(0-9)/(-2*14.85)$

$= 0.30 m$ $=0.30m$

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A box with an initial speed of $3 \frac{m}{s}$ $3 m/s$ is moving up a ramp. The ramp has a kinetic friction coefficient of $\frac{8}{7}$ $8/7$ and an incline of $\frac{π}{4}$ $( pi )/4$ . 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 $3 \frac{m}{s}$ $3 m/s$ is moving up a ramp. The ramp has a kinetic friction coefficient of $\frac{8}{7}$ $8/7$ and an incline of $\frac{π}{4}$ $( pi )/4$ . How far along the ramp will the box go?