# If a #5 kg# object moving at #6 m/s# slows to a halt after moving #2 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 coefficient of kinetic friction is

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To find the coefficient of kinetic friction, we can use the equation:

[ F_{\text{friction}} = \mu_k \times F_{\text{normal}} ]

Where:

- ( F_{\text{friction}} ) is the force of friction,
- ( \mu_k ) is the coefficient of kinetic friction,
- ( F_{\text{normal}} ) is the normal force.

First, we need to find the force of friction using Newton's second law:

[ F_{\text{friction}} = m \times a ]

Where:

- ( m ) is the mass of the object (5 kg),
- ( a ) is the acceleration.

We know the object slows down to a halt, so the final velocity is 0 m/s. Using the equation of motion:

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

Where:

- ( v ) is the final velocity (0 m/s),
- ( u ) is the initial velocity (6 m/s),
- ( a ) is the acceleration,
- ( s ) is the displacement (2 m).

Rearranging the equation to solve for acceleration:

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

[ a = \frac{{0^2 - 6^2}}{{2 \times 2}} = -9 , \text{m/s}^2 ]

Now, using Newton's second law:

[ F_{\text{friction}} = m \times a = 5 , \text{kg} \times (-9 , \text{m/s}^2) = -45 , \text{N} ]

Since the object is slowing down, the force of friction acts in the direction opposite to the motion, hence the negative sign.

The normal force can be calculated as:

[ F_{\text{normal}} = m \times g ]

Where:

- ( g ) is the acceleration due to gravity (9.8 m/s²).

[ F_{\text{normal}} = 5 , \text{kg} \times 9.8 , \text{m/s}^2 = 49 , \text{N} ]

Now, we can find the coefficient of kinetic friction:

[ \mu_k = \frac{{F_{\text{friction}}}}{{F_{\text{normal}}}} ]

[ \mu_k = \frac{{-45 , \text{N}}}{{49 , \text{N}}} ]

[ \mu_k \approx -0.92 ]

However, coefficients of friction cannot be negative, so we take the absolute value:

[ \mu_k \approx 0.92 ]

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