# A box with an initial speed of #3 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #1/6 # and an incline of #( pi )/3 #. How far along the ramp will the box go?

The distance is

Consequently, the object's net force is

Newton's Second Law of Motion states

So

A deceleration is indicated by the negative sign.

Utilize the equation of motion.

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To find the distance along the ramp the box will go, we can use the equation for the work done by friction:

(W_{friction} = \mu_k \cdot m \cdot g \cdot d)

Given:

- Initial speed (v_i = 3 , \text{m/s})
- Kinetic friction coefficient (\mu_k = \frac{1}{6})
- Incline angle (\theta = \frac{\pi}{3})

First, we find the acceleration of the box along the incline using the kinematic equation:

(v_f^2 = v_i^2 + 2 \cdot a \cdot d)

Solving for acceleration (a):

(a = \frac{v_f^2 - v_i^2}{2 \cdot d})

Since the box is moving up the ramp, the final velocity (v_f = 0), so:

(a = \frac{-v_i^2}{2 \cdot d})

Next, we find the gravitational force component parallel to the ramp:

(F_{\text{gravity, parallel}} = m \cdot g \cdot \sin(\theta))

The force of friction opposing motion is:

(F_{\text{friction}} = \mu_k \cdot m \cdot g \cdot \cos(\theta))

At equilibrium, the net force parallel to the ramp is zero:

(F_{\text{gravity, parallel}} - F_{\text{friction}} = 0)

Substitute the expressions for these forces and solve for (d):

(m \cdot g \cdot \sin(\theta) - \mu_k \cdot m \cdot g \cdot \cos(\theta) = 0)

(d = \frac{\mu_k}{\tan(\theta)})

Now, substitute the given values:

(d = \frac{\frac{1}{6}}{\tan\left(\frac{\pi}{3}\right)})

(d \approx \frac{\frac{1}{6}}{\sqrt{3}})

(d \approx \frac{1}{6\sqrt{3}})

(d \approx \frac{\sqrt{3}}{18} , \text{m})

So, the box will travel approximately (\frac{\sqrt{3}}{18} , \text{m}) along the ramp.

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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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