# A box with an initial speed of #7 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #2/5 # and an incline of #pi /12 #. How far along the ramp will the box go?

The distance is

Then the net force on the object is

According to Newton's Second Law of Motion

So

The negative sign indicates a deceleration

Apply the equation of motion

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First, calculate the acceleration due to gravity acting parallel to the ramp, which equals ( g \cdot \sin(\pi/12) ). Then, calculate the net force parallel to the ramp by subtracting the force of friction from the component of the gravitational force parallel to the ramp. Use Newton's second law to find the acceleration of the box along the ramp. Finally, use the kinematic equation ( v^2 = u^2 + 2as ) to find the distance ( s ) traveled by the box, where ( u ) is the initial velocity (7 m/s), ( v ) is the final velocity (0 m/s at the top), and ( a ) is the acceleration found earlier. Solve for ( s ) to find the distance the box travels 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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