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

Question

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

Emily Abbate · Accepted Answer

The distance is #=5.3m#

Taking the direction up and parallel to the plane as positive #↗^+#

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

Consequently, the object's net force is

#F=-F_r-Wsintheta#

#=-F_r-mgsintheta#

#=-mu_kN-mgsintheta#

#=mmu_kgcostheta-mgsintheta#

Newton's Second Law states

#F=m*a#

Where #a# is the acceleration

So

#ma=-mu_kgcostheta-mgsintheta#

#a=-g(mu_kcostheta+sintheta)#

The coefficient of kinetic friction is #mu_k=1/2#

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

The incline of the ramp is #theta=2/3pi#

#a=-9.8*(1/2cos(2/3pi)+sin(2/3pi))#

#=-6.04ms^-2#

A deceleration is indicated by the negative sign.

We utilize the equation of motion.

#v^2=u^2+2as#

#u=8ms^-1#

#v=0#

#a=-6.04ms^-2#

#s=(v^2-u^2)/(2a)#

#=(0-64)/(-2*6.04)#

#=5.3m#

Nora Arzate · Answer

undefined To find the distance along the ramp that the box will go, you can use the kinematic equations of motion along with the equations for friction and the forces acting on the box. First, calculate the net force acting on the box along the ramp, then use this force to find the acceleration of the box. Finally, use the equations of motion to find the distance traveled by the box along the ramp.