water skiers often ride to one side of the center line of a boat, as shown 7-14. In this case, the ski boat is traveling at 15m/s and the tension in the rope is 80 n. If the boat does 3400 J of work on the skier in 50.3 m, what is the angle (theta) between the tow rope and the center line of the boat?
To find the angle (theta) between the tow rope and the center line of the boat, we can use the work-energy principle.
The work done on an object is given by the equation:
Work = Force * Distance * Cos(theta)
In this case, the work done on the skier is given as 3400 J (joules), the force is the tension in the rope (80 N), and the distance is the distance the skier is pulled (50.3 m).
Plugging the given values into the formula, we have:
3400 J = 80 N * 50.3 m * Cos(theta)
To solve for theta, we need to rearrange the formula. Dividing both sides of the equation by (80 N * 50.3 m), we get:
3400 J / (80 N * 50.3 m) = Cos(theta)
Simplifying the left side of the equation, we have:
0.848 = Cos(theta)
Now, to find theta, we need to take the inverse cosine (also known as arccos) of both sides of the equation:
theta = arccos(0.848)
Using a calculator or a trigonometric table, you can find that the value of theta is approximately 31.3 degrees.
Therefore, the angle (theta) between the tow rope and the center line of the boat is approximately 31.3 degrees.