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?

To find the final speed of the crate, we can use the principle of work and energy. The frictionless portion of the floor doesn't affect the final speed, so we only need to consider the portion of the floor with friction.

First, let's calculate the work done by the constant horizontal force on the crate on the portion of the floor with friction. The work done by a force is equal to the force multiplied by the displacement and the cosine of the angle between the force and displacement vectors. In this case, the angle is 0 degrees since the force and displacement are parallel.

Work done on the crate = Force * displacement * cos(angle)

The force is 400 N and the displacement is 17 m, so the work done on the crate is:

Work done on the crate = 400 N * 17 m * cos(0)

Since the cosine of 0 degrees is 1, the work done on the crate is:

Work done on the crate = 400 N * 17 m * 1 = 6800 J

Next, let's calculate the work done against friction on the crate. The work done against friction is equal to the force of friction multiplied by the displacement.

The force of friction is given by 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 the formula:

Weight = mass * gravitational acceleration

The mass of the crate is 120 kg and the gravitational acceleration is approximately 9.8 m/s^2, so the weight of the crate is:

Weight = 120 kg * 9.8 m/s^2 = 1176 N

Substituting this into the equation for the force of friction:

Force of friction = 0.34 * 1176 N = 399.84 N (approximately 400 N)

The work done against friction is equal to the force of friction multiplied by the displacement of 17 m:

Work done against friction = 400 N * 17 m = 6800 J

Since the work done by the constant horizontal force is equal to the work done against friction, the total work done on the crate is:

Total work done on the crate = Work done on the crate + Work done against friction

Total work done on the crate = 6800 J + 6800 J = 13600 J

According to the work-energy principle, the work done on an object is equal to the change in its kinetic energy. Therefore, the total work done on the crate is equal to the change in its kinetic energy:

Total work done on the crate = Change in kinetic energy

The initial kinetic energy of the crate is zero because it starts from rest. The final kinetic energy of the crate can be calculated using the formula:

Kinetic energy = 0.5 * mass * velocity^2

Since we're looking for the final speed of the crate, we can rewrite the formula as:

Final speed = square root of (2 * (total work done on the crate) / mass)

Substituting the values, we have:

Final speed = square root of (2 * 13600 J / 120 kg)

Final speed = square root of (27200 J / 120 kg)

Final speed = square root of 226.67 m^2/s^2

Final speed = 15.05 m/s (rounded to two decimal places)

Therefore, the final speed of the crate is approximately 15.05 m/s.