An object glides on a horizontal tabletop with a coefficient of kinetic friction of 0.55. If its initial velocity is 4.1 m/s, how long does it take for the object to come to rest?

F = m a

-.55 * m * g = m * a

a = -.55 g = -5.39

v = Vi + a t
0 = 4.1 -5.39 t

t = 4.1/5.39

To calculate how long it takes for the object to come to rest, we first need to find the acceleration due to friction. We can use the formula:

Frictional force = coefficient of friction * normal force

Since the object is on a horizontal tabletop and not accelerating vertically, the normal force is equal to the weight of the object, given by:

Normal force = mass * acceleration due to gravity

Now, the frictional force is also equal to the mass of the object multiplied by its acceleration due to friction, given by:

Frictional force = mass * acceleration due to friction

Since the object is coming to rest, its acceleration due to friction is equal to its deceleration. Therefore, we can equate the frictional force and the mass multiplied by deceleration:

mass * acceleration due to friction = mass * deceleration

The mass cancels out, so we have:

acceleration due to friction = deceleration

We can use the formula for deceleration:

acceleration = (final velocity - initial velocity) / time

Since the object comes to rest, the final velocity is 0. Therefore, we have:

acceleration due to friction = (-initial velocity) / time

Now, we can solve for time:

time = (-initial velocity) / (acceleration due to friction)

Using the given values, we can substitute in:

time = (-4.1 m/s) / (acceleration due to friction)

Given that the coefficient of kinetic friction is 0.55, we can substitute this value into the equation:

time = (-4.1 m/s) / (0.55)

Simplifying the equation:

time = -7.45 s

Since time cannot be negative, we take the absolute value:

time = 7.45 s

Therefore, it takes approximately 7.45 seconds for the object to come to rest.

To find the time it takes for the object to come to rest, we need to use the laws of motion and consider the forces acting on the object.

First, we need to determine the force of kinetic friction that acts on the object. The force of kinetic friction (Fk) is given by the equation:

Fk = μk * N

where μk is the coefficient of kinetic friction and N is the normal force. In this case, since the object is on a horizontal tabletop, the normal force is equal to the weight of the object, which can be calculated using the equation:

N = m * g

where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Next, we need to determine the acceleration (a) of the object using Newton's second law of motion, which states that the net force acting on an object is equal to the product of mass and acceleration:

Fnet = m * a

In this case, the net force acting on the object is the force of kinetic friction (Fk) in the opposite direction of the object's velocity. So,

Fk = m * a

Finally, we can use the equations of motion to find the time (t) it takes for the object to come to rest. The equation to use is:

v = u + a*t

where v is the final velocity (which is 0 m/s since the object comes to rest), u is the initial velocity, a is the acceleration (calculated from the force of kinetic friction), and t is the time.

Let's plug in the given values and solve for t:

Given:
Coefficient of kinetic friction (μk) = 0.55
Initial velocity (u) = 4.1 m/s

1. Calculate the force of kinetic friction:
N = m * g
Fk = μk * N

2. Calculate the acceleration:
Fk = m * a

3. Calculate the time:
v = u + a*t

By following these steps, you can calculate the time it takes for the object to come to rest.