A box with an initial speed of #9 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #5/6 # and an incline of #(3 pi )/8 #. 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 how far along the ramp the box will go, you can use the equations of motion. First, calculate the acceleration of the box along the ramp using the gravitational force component parallel to the ramp and subtracting the frictional force. Then, use the kinematic equation to find the distance traveled. Given the incline angle, you can decompose the gravitational force into components parallel and perpendicular to the ramp. The parallel component provides the acceleration. Use the equation:
[ a = g \cdot \sin(\theta) - \mu_k \cdot g \cdot \cos(\theta) ]
where:
- ( a ) = acceleration
- ( g ) = acceleration due to gravity (approximately ( 9.8 , \text{m/s}^2 ))
- ( \theta ) = incline angle
- ( \mu_k ) = coefficient of kinetic friction
Then, use the equation of motion:
[ d = v_i \cdot t + \frac{1}{2} \cdot a \cdot t^2 ]
where:
- ( d ) = distance traveled
- ( v_i ) = initial velocity
- ( t ) = time
Given:
- ( v_i = 9 , \text{m/s} )
- ( \theta = \frac{3\pi}{8} )
- ( \mu_k = \frac{5}{6} )
Substitute the values and solve for ( d ).
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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.
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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