A box with an initial speed of #7 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #5/6 # and an incline of #pi /4 #. How far along the ramp will the box go?

Answer 1

The distance is #=1.93m#

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
#ma=-mu_kgcostheta-mgsintheta#
#a=-g(mu_kcostheta+sintheta)#
The coefficient of kinetic friction is #mu_k=5/6#
The incline of the ramp is #theta=1/6pi#
#a=-9.8*(5/6cos(1/4pi)+sin(1/4pi))#
#=-12.7ms^-2#

A deceleration is indicated by the negative sign.

We utilize the equation of motion.

#v^2=u^2+2as#
#u=7ms^-1#
#v=0#
#a=-12.7ms^-2#
#s=(v^2-u^2)/(2a)#
#=(0-49)/(-2*12.7)#
#=1.93m#
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Answer 2

To find the distance along the ramp the box will go, you can use the following steps:

  1. Calculate the acceleration of the box along the ramp due to gravity.
  2. Calculate the frictional force acting against the motion of the box.
  3. Calculate the net force acting on the box along the ramp.
  4. Use the net force to find the acceleration of the box along the ramp.
  5. Use the kinematic equation to find the distance along the ramp the box will go.

Let's start by finding the acceleration of the box along the ramp due to gravity:

Acceleration due to gravity along the ramp = g * sin(π/4)

Next, calculate the frictional force acting against the motion of the box:

Frictional force = coefficient of kinetic friction * normal force

Normal force = weight of the box * cos(π/4)

Now, calculate the net force acting on the box along the ramp:

Net force = gravitational force along the ramp - frictional force

Use the net force to find the acceleration of the box along the ramp:

Acceleration along the ramp = Net force / mass of the box

Finally, use the kinematic equation to find the distance along the ramp the box will go:

Distance along the ramp = (initial velocity * time) + (0.5 * acceleration along the ramp * time^2)

where time can be found using the equation: time = final velocity / acceleration along the ramp, and the final velocity is 0 m/s since the box comes to rest.

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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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