A box with an initial speed of #6 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

The distance is #=1.56m#

Resolving in 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 of Motion states

#F=m*a#
Where #a# is the acceleration of the box

So

#ma=-mu_kgcostheta-mgsintheta#
#a=-g(mu_kcostheta+sintheta)#
The coefficient of kinetic friction is #mu_k=2/3#
The acceleration due to gravity is #g=9.8ms^-2#
The incline of the ramp is #theta=3/8pi#
The acceleration is #a=-9.8*(2/3cos(3/8pi)+sin(3/8pi))#
#=-11.55ms^-2#

A deceleration is indicated by the negative sign.

Utilize the equation of motion.

#v^2=u^2+2as#
The initial velocity is #u=6ms^-1#
The final velocity is #v=0#
The acceleration is #a=-11.55ms^-2#
The distance is #s=(v^2-u^2)/(2a)#
#=(0-36)/(-2*11.55)#
#=1.56m#
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Answer 2

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

  1. Calculate the gravitational force component parallel to the ramp: ( F_{\text{gravity}} = m \cdot g \cdot \sin(\theta) )

  2. Determine the frictional force opposing the motion: ( F_{\text{friction}} = \mu_k \cdot N )

  3. Find the net force parallel to the ramp: ( F_{\text{net}} = F_{\text{gravity}} - F_{\text{friction}} )

  4. Use Newton's second law to find the acceleration: ( a = \frac{F_{\text{net}}}{m} )

  5. Use the kinematic equation to find the distance traveled: ( d = \frac{v_i^2}{2a} )

Where:

  • ( m ) is the mass of the box,
  • ( g ) is the acceleration due to gravity,
  • ( \theta ) is the angle of incline,
  • ( \mu_k ) is the coefficient of kinetic friction,
  • ( N ) is the normal force exerted by the ramp on the box,
  • ( v_i ) is the initial velocity of the box,
  • ( a ) is the acceleration of the box, and
  • ( d ) is the distance traveled by the box 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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