# A box with an initial speed of #4 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.

We utilize the equation of motion.

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The distance along the ramp the box will go is approximately 9.44 meters.

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To determine how far along the ramp the box will go, we can use the principles of physics, specifically those related to motion along inclined planes.

First, we need to calculate the net force acting on the box along the ramp. This force is composed of two components: the force due to gravity acting down the incline and the force of kinetic friction opposing the motion.

The force due to gravity acting down the incline can be calculated using the formula:

[ F_{\text{gravity}} = m \cdot g \cdot \sin(\theta) ]

where:

- ( m ) is the mass of the box,
- ( g ) is the acceleration due to gravity (approximately ( 9.8 , \text{m/s}^2 )), and
- ( \theta ) is the angle of the incline.

The force of kinetic friction opposing the motion can be calculated using the formula:

[ F_{\text{friction}} = \mu_k \cdot N ]

where:

- ( \mu_k ) is the coefficient of kinetic friction, and
- ( N ) is the normal force acting on the box, which can be calculated as:

[ N = m \cdot g \cdot \cos(\theta) ]

Once we have both forces, we can find the net force by subtracting the force of friction from the force due to gravity:

[ F_{\text{net}} = F_{\text{gravity}} - F_{\text{friction}} ]

Next, we use Newton's second law of motion, which states that the net force acting on an object is equal to its mass times its acceleration (( F_{\text{net}} = m \cdot a )). Since we are interested in the distance traveled, we use the formula for distance traveled along an incline with constant acceleration:

[ d = \frac{v_i^2}{2a} ]

where:

- ( d ) is the distance traveled,
- ( v_i ) is the initial velocity (given as ( 4 , \text{m/s} )), and
- ( a ) is the acceleration, which we calculate from Newton's second law.

By substituting the known values and solving the equations, we can find the distance ( d ) along the ramp that the box will travel.

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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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