A water-skier is being pulled by a tow rope attached to a boat. As the driver pushes the throttle forward, the skier accelerates. A 83.0-kg water-skier has an initial speed of 5.9 m/s. Later, the speed increases to 11.4 m/s. Determine the work done by the net external force acting on the skier.
Well, we can get the change in kinetic energy which is the net work done. This is greatly simplified because most of the actual (not net) work done by the tow boat is wasted in wave and frictional drag: (Thrust - Drag = net force = m a)
Work = (1/2)(83)(11.4^2 - 5.9^2)
To determine the work done by the net external force acting on the skier, we can use the work-energy principle, which states that the net work done on an object is equal to its change in kinetic energy.
The formula for work is given by:
Work = force * displacement * cos(theta)
In this case, the net external force acting on the skier is acting in the direction of the displacement, so we can simplify the formula to:
Work = force * displacement
We can calculate the force using Newton's second law:
force = mass * acceleration
In this case, the mass of the water-skier is 83.0 kg, and we are given the initial and final speeds. We can calculate the acceleration using the following formula:
acceleration = (final velocity^2 - initial velocity^2) / (2 * displacement)
We know that the mass, initial velocity, and final velocity are given. We need to determine the displacement of the skier in order to calculate the work done.
Using the formula for average velocity:
average velocity = (initial velocity + final velocity) / 2
And rearranging the formula to solve for displacement:
displacement = (final velocity - initial velocity) * (2 * average velocity)
Plugging in the values, we can now calculate the displacement:
displacement = (11.4 m/s - 5.9 m/s) * (2 * ((5.9 m/s + 11.4 m/s) / 2))
displacement = 5.5 m/s * (2 * 8.65 m/s)
displacement = 5.5 m/s * 17.3 m/s
displacement = 95.15 m²/s²
Now that we have the displacement, we can calculate the acceleration:
acceleration = (11.4 m/s² - 5.9 m/s²) / (2 * 95.15 m²/s²)
acceleration = 5.5 m/s² / 190.3 m²/s²
acceleration = 0.0289 m/s²
Finally, we can calculate the force:
force = 83.0 kg * 0.0289 m/s²
force = 2.397 kg·m/s²
Now that we have the force, we can calculate the work done:
work = force * displacement
work = 2.397 kg·m/s² * 2 * 8.65 m/s
work = 41.3149 kg·m²/s²
Therefore, the work done by the net external force acting on the skier is approximately 41.31 kg·m²/s².