Final answer:
To calculate the negative work needed to slow down the barge, we use the work-energy principle. Using the given mass, initial velocity, and final velocity, we calculate the force. Then, assuming a time of 1 second, we find the distance over which the force acts. Substituting the values into the work formula, we can calculate the negative work in joules.
Step-by-step explanation:
To calculate the negative work needed to slow down the barge, we can use the work-energy principle, which states that the work done on an object equals the change in its kinetic energy. The work formula is given as:
Work = Force * Distance * cos(θ)
First, we need to find the force acting on the barge. The force can be calculated using Newton's second law:
Force = Mass * Acceleration
Since the barge is slowing down, its acceleration is in the opposite direction of its velocity. So, we can take the magnitude of the acceleration as negative. Given that the mass of the barge is 588 kg, the initial velocity is 1.1 m/s, and the final velocity is 0.4 m/s, we can calculate the force:
Force = 588 kg * (-1.1 m/s - 0.4 m/s)
Next, we need to find the distance over which the force acts. Since it's not specified in the question, we can assume that the distance is the same as the displacement of the barge. Using the equation:
Distance = (Final Velocity - Initial Velocity) * time
Assuming the time is 1 second:
Distance = (0.4 m/s - 1.1 m/s) * 1 s
Finally, we can substitute the values into the work formula:
Work = (588 kg * (-1.1 m/s - 0.4 m/s)) * ((0.4 m/s - 1.1 m/s) * 1 s) * cos(θ)
Simplifying the equation, we can calculate the negative work in joules.