A 120-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 400 N. For the first 17 m the floor is frictionless, and for the next 17 m the coefficient of friction is 0.34. What is the final speed of the crate?

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What does the "u" stand for?

ex.... m = mass

coefficent of friction. Normally we abreviate it with "mu", the sound of the Greek letter that looks like a u.

To find the final speed of the crate, we need to calculate the acceleration of the crate in the two different situations (frictionless and with friction).

First, let's calculate the acceleration when the floor is frictionless. In this case, there is no friction acting on the crate. Therefore, the only force acting on the crate is the applied force of 400 N.

We can use Newton's second law of motion, which states that the acceleration of an object is equal to the net force applied on it divided by its mass:

acceleration = (net force) / (mass)

In this case, the net force is equal to the applied force:

acceleration(frictionless) = 400 N / 120 kg
acceleration(frictionless) = 3.33 m/s²

Now, let's consider the situation when the coefficient of friction is 0.34. In this case, the force of friction needs to be calculated and subtracted from the applied force to find the net force.

The force of friction can be found using the equation:

force of friction = coefficient of friction * normal force

The normal force is equal to the weight of the crate, which can be calculated using:

normal force = mass * gravity

The acceleration with friction can be calculated using the net force divided by the mass:

acceleration(friction) = (applied force - force of friction) / mass

Let's calculate the force of friction and the acceleration:

normal force = 120 kg * 9.8 m/s²
normal force = 1176 N

force of friction = 0.34 * 1176 N
force of friction = 399.84 N

acceleration(friction) = (400 N - 399.84 N) / 120 kg
acceleration(friction) = 0.0013 m/s²

Next, we need to calculate the time taken to travel each section of the floor.

The time taken when the floor is frictionless can be calculated using the equation:

time(frictionless) = sqrt((2 * displacement) / acceleration(frictionless))

time(frictionless) = sqrt((2 * 17 m) / 3.33 m/s²)
time(frictionless) = 2.56 s

The time taken when the floor has friction can be calculated using the equation:

time(friction) = sqrt((2 * displacement) / acceleration(friction))

time(friction) = sqrt((2 * 17 m) / 0.0013 m/s²)
time(friction) = 258.2 s

Finally, to find the final speed of the crate, we need to calculate the total displacement and the average velocity during the two stages.

The total displacement is the sum of the displacements in each stage:

total displacement = 17 m + 17 m
total displacement = 34 m

The average velocity is the total displacement divided by the total time taken:

average velocity = total displacement / (time(frictionless) + time(friction))

average velocity = 34 m / (2.56 s + 258.2 s)
average velocity = 0.1311 m/s

Therefore, the final speed of the crate is approximately 0.1311 m/s.