A sled is pushed by 12 monkeys with a constant force until it reaches a velocity of 18.9 m/s. All 12 monkeys stop pushing and the sled slides across a rough surface. From the time the monkeys stop pushing the box slides 12.3 meters before it comes to a stop. If the u of k of the contact is 0.21, what is the mass of the sled?

To find the mass of the sled, we can use the concept of work and energy.

First, let's determine the work done by the monkeys to accelerate the sled to its final velocity of 18.9 m/s. The work is given by the equation:

Work = Force × Distance × cosθ

Since the force is constant, we can find the work done by one monkey and then multiply it by 12 to account for all 12 monkeys. To find the work done by one monkey, we need to know the angle θ between the direction of the force and the direction of motion. Let's assume that the angle θ is 0°, so cosθ = 1.

Now let's calculate the work done. Since the sled is already in motion, the work done by the monkeys goes into increasing its kinetic energy.

Kinetic energy = (1/2) × Mass × Velocity^2

The work done by the monkeys is equal to the change in kinetic energy:

Work = ΔKinetic energy

Therefore,

Force × Distance = (1/2) × Mass × (Final Velocity^2 - Initial Velocity^2)

Substituting the known values:

12 × Force × Distance = (1/2) × Mass × (18.9^2 - 0^2)

Next, let's calculate the force using the equation:

Force = Mass × Acceleration

Since the force is constant, we can rearrange the equation to solve for force:

Force = Mass × (Final Velocity - Initial Velocity)/(time)

In this case, the initial velocity is 0 m/s, the final velocity is 18.9 m/s, and the time is not provided. Therefore, we need to find the time it took for the sled to reach 18.9 m/s.

We can use the equation of motion:

Final Velocity = Initial Velocity + (Acceleration × Time)

Since the sled starts from rest, the initial velocity is 0 m/s. Rearranging the equation, we have:

Time = Final Velocity / Acceleration

Now, let's calculate the time:

Time = 18.9 m/s / Acceleration

The acceleration can be found using the equation:

Acceleration = Force / Mass

Substituting this value of the acceleration into the equation for time, we get:

Time = 18.9 m/s / (Force / Mass)

Substituting the expression for force from earlier, we have:

Time = 18.9 m/s / (Mass × (Final Velocity - Initial Velocity) / (12 × Distance))

Now that we have the time, we can go back to the equation for force and solve for it:

Force = Mass × (Final Velocity - Initial Velocity) / Time

Substituting the values:

Force = Mass × (18.9 m/s - 0 m/s) / Time

Finally, we can substitute the force back into the equation for work to solve for the mass:

12 × Force × Distance = (1/2) × Mass × (18.9^2 - 0^2)

Rearranging the equation, we have:

Mass = (12 × Force × Distance) / ((1/2) × (18.9^2))

Now, substituting the known values:

Mass = (12 × Force × Distance) / (0.5 × 18.9^2)

Lastly, you stated that the coefficient of kinetic friction is given as 0.21. Using this coefficient and the normal force, we can calculate the force of friction:

Force of friction = Coefficient of friction × Normal force

The normal force is equal to the weight of the sled, which is given by:

Normal force = Mass × Gravity

Finally, we can substitute the force of friction back into the equation for force to get the mass of the sled.