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?
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 the box will travel along the ramp, you can use the following steps:

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

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

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

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

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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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 A truck pulls boxes up an incline plane. The truck can exert a maximum force of #3,500 N#. If the plane's incline is #(5 pi )/12 # and the coefficient of friction is #5/12 #, what is the maximum mass that can be pulled up at one time?
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 If an object is moving at #22 m/s# over a surface with a kinetic friction coefficient of #u_k=11 /g#, how far will the object continue to move?
 An object with a mass of #2 kg# is on a surface with a kinetic friction coefficient of # 7 #. How much force is necessary to accelerate the object horizontally at #28 m/s^2#?
 An object with a mass of #5 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= e^x2x+3 #. How much work would it take to move the object over #x in [3, 4], where x is in meters?
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