A warehouse worker exerts a constant horizontal force of magnitude 89.0 N on a 43.0 kg box that is initially at rest on the horizontal floor of the warehouse. When the box has moved a distance of 1.60 m, its speed is 1.10 m/s. What is the coefficient of kinetic friction between the box and the floor?

To find the coefficient of kinetic friction, we need to analyze the forces acting on the box.

First, let's calculate the net horizontal force exerted on the box. We know that force is equal to mass multiplied by acceleration.

F_net = m * a

Since the box is initially at rest and then moves with a certain speed, it undergoes acceleration. We can use the kinematic equation to find the acceleration:

v^2 = u^2 + 2a * s

where v is the final velocity (1.10 m/s), u is the initial velocity (0 since the box is at rest), a is the acceleration (which is constant), and s is the distance traveled (1.60 m).

By plugging in the values, we can solve for the acceleration:

1.10^2 = 0 + 2 * a * 1.60
1.21 = 3.2 * a
a = 1.21 / 3.2
a = 0.3781 m/s^2

Now, let's consider the forces acting on the box. There are three forces involved:

1. The horizontal force exerted by the worker.
2. The force of kinetic friction between the box and the floor.
3. The normal force exerted by the floor on the box in the vertical direction.

The net force exerted on the box is the result of these forces:

F_net = F_applied - F_friction

We can calculate the applied force using the formula given in the problem:

F_applied = 89.0 N

The friction force can be calculated using:

F_friction = μ * N

where μ is the coefficient of kinetic friction and N is the normal force.

From Newton's second law, we know that the net force is equal to mass multiplied by acceleration:

F_net = m * a

Substituting the values and solving for the normal force:

F_applied - F_friction = m * a
89.0 N - μ * N = (43.0 kg) * (0.3781 m/s^2)

Simplifying the equation:

89.0 - μ * N = 16.2443

Now, let's consider the normal force. When a box is on a horizontal surface, the normal force is equal to the weight of the box, which is given by:

N = m * g

where g is the acceleration due to gravity (9.8 m/s^2).

Substituting the value:

N = (43.0 kg) * (9.8 m/s^2)
N = 421.4 N

Plugging this into the equation:

89.0 - μ * 421.4 = 16.2443

Simplifying:

-μ * 421.4 = 16.2443 - 89.0
-μ * 421.4 = -72.7557

Now, we can solve for the coefficient of kinetic friction, μ:

μ = (-72.7557) / (421.4)
μ ≈ -0.173

Since the coefficient of kinetic friction cannot be negative, we take the absolute value:

μ ≈ 0.173

Therefore, the coefficient of kinetic friction between the box and the floor is approximately 0.173.