A 82.0 kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 314.0 N. The floor is rough with a coefficient of friction of 0.30. What is the final speed of the crate, in m/s, after it has been pulled a distance of 13.0 m?

M*g = 82 * 9.8 = 803.6 N. = Wt. of crate. = Normal force, Fn.

Fk = u*Fn = 0.3 * 803.6 = 241 N. = Force
of kinetic friction.

F-Fk = M*a.
314-241 = 82*a.
82a = 73.
a = 0.890 m/s^2.

Vf^2 = Vo^2 + 2a*d.
Vo = 0.
Vf = ?.

To determine the final speed of the crate, we need to calculate the net force acting on it and then use Newton's second law of motion to find the acceleration. Finally, we can use the kinematic equation to calculate the final speed.

Step 1: Calculate the force of friction:
The force of friction can be calculated using the coefficient of friction (μ) and the normal force (N). In this case, the normal force is equal to the weight of the crate, which is the mass (m) multiplied by the acceleration due to gravity (g ≈ 9.8 m/s^2).

Normal force (N) = mass (m) * acceleration due to gravity (g)
N = 82.0 kg * 9.8 m/s^2

Next, we can calculate the force of friction (F_friction) using the formula:

F_friction = coefficient of friction (μ) * normal force (N)
F_friction = 0.30 * (82.0 kg * 9.8 m/s^2)

Step 2: Calculate the net force:
The net force acting on the crate is the horizontal force applied (F_applied) minus the force of friction (F_friction).

Net force (F_net) = F_applied - F_friction
F_net = 314.0 N - F_friction (calculated in step 1)

Step 3: Calculate the acceleration:
Using Newton's second law of motion, we can calculate the acceleration (a) by dividing the net force (F_net) by the mass (m) of the crate.

Acceleration (a) = Net force (F_net) / mass (m)
a = F_net / 82.0 kg

Step 4: Calculate the final speed:
We can use the kinematic equation to calculate the final speed of the crate. The initial velocity (v_0) is 0 since the crate starts from rest, and the distance traveled (d) is given as 13.0 m.

Final speed (v) can be calculated using the equation:
v^2 = v_0^2 + 2 * a * d

Given that v_0 = 0, we can simplify the equation to:
v^2 = 2 * a * d

Finally, we can solve for the final speed (v) by taking the square root of both sides of the equation.

v = sqrt(2 * a * d)

Now, you can substitute the values you have calculated into this equation to find the final speed of the crate.