# If a #4 kg# object moving at #5/2 m/s# slows to a halt after moving #12 m#, what is the coefficient of kinetic friction of the surface that the object was moving over?

The coefficient of kinetic friction is

Apply the equation of motion

To calculate the acceleration

The acceleration is

According to Newton's Second Law, the force of friction is

The normal force is

The coefficient of kinetic friction is

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The coefficient of kinetic friction can be calculated using the equation:

[ \text{Frictional Force} = \text{Coefficient of Kinetic Friction} \times \text{Normal Force} ]

The frictional force can also be expressed as the product of the coefficient of kinetic friction and the normal force. In this case, since the object comes to a halt, the frictional force is equal to the force that caused the deceleration. The force that caused the deceleration can be calculated using Newton's second law:

[ F = m \times a ]

Where ( m ) is the mass of the object and ( a ) is the acceleration.

To find the acceleration, we can use the equation of motion:

[ v^2 = u^2 + 2as ]

Where:

- ( v ) is the final velocity (which is 0 m/s since the object comes to a halt),
- ( u ) is the initial velocity,
- ( a ) is the acceleration,
- ( s ) is the displacement.

Rearranging the equation, we have:

[ a = \frac{{v^2 - u^2}}{{2s}} ]

Now, we can substitute the given values:

- ( m = 4 ) kg
- ( u = \frac{5}{2} ) m/s
- ( v = 0 ) m/s
- ( s = 12 ) m

Calculate the acceleration using the equation:

[ a = \frac{{(0)^2 - \left(\frac{5}{2}\right)^2}}{{2 \times 12}} ]

Once we find the acceleration, we can use Newton's second law to find the force of friction. Then, we can find the normal force using the equation:

[ \text{Normal Force} = m \times g ]

Where ( g ) is the acceleration due to gravity (( 9.8 , \text{m/s}^2 )).

Finally, substitute the values into the equation for the coefficient of kinetic friction:

[ \text{Coefficient of Kinetic Friction} = \frac{{\text{Frictional Force}}}{{\text{Normal Force}}} ]

Calculate the coefficient of kinetic friction using the values obtained.

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