Tarzan, King of the Jungle (mass \(=70.4 \mathrm{~kg}\) ), grabs a vine of length \(14.5 \mathrm{~m}\) hanging from a tree branch. The angle of the vine was \(25.9^{\circ}\) with respect to the vertical when he grabbed it. At the lowest point of his trajectory, he picks up Jane (mass \(=43.4 \mathrm{~kg}\) ) and continues his swinging motion. What angle relative to the vertical will the vine have when Tarzan and Jane reach the highest point of their trajectory?

Kinetic energy is the energy an object possesses due to its motion. It's a vital concept in mechanics and is usually denoted by the symbol KE. The formula for kinetic energy of a moving object is given by
\( KE = \frac{1}{2}mv^2 \)
where m is the object's mass and v is its velocity. In the Tarzan and Jane problem, when Tarzan is at the lowest point of his swing, all his initial potential energy from being at a height is converted into kinetic energy. As there is no other energy input or dissipation, the initial potential energy and kinetic energy at the lowest point are equal in magnitude. This conservation of energy allows us to set up an equation that equates the two and solve for unknown variables.

Potential energy is the stored energy an object has due to its position or state. In the context of our problem, gravitational potential energy is most relevant, which is dependent on the object's height above a reference point, its mass, and the acceleration due to gravity (g). The formula for gravitational potential energy (PE) is:
\( PE = mgh \)
where m is mass, g is acceleration due to gravity, and h is the height above the reference point. In the step-by-step solution, potential energy calculations establish the energy states at different points in Tarzan's swing. Understanding potential energy is crucial for solving problems involving height and gravity.

Trigonometry, the branch of mathematics dealing with the relationships between the sides and angles of triangles, is essential in physics, particularly in mechanics. It provides the tools to model and solve problems involving angles, heights, and distances. In the Tarzan problem, trigonometry helps us determine the change in height as he swings by using the vine's length and the angle made with the vertical. The basic trigonometric functions sin, cos, and tan are often used in such calculations. Specifically, the original exercise uses the cosine function to relate the angle of the vine to the height of Tarzan above the lowest point of the swing, allowing us to calculate the initial and final potential energies.

Mechanics is the branch of physics dealing with the motion of objects and the forces causing those motions. It typically involves analyzing situations to determine the net effect of forces and applying conservation laws, such as conservation of energy and momentum, to predict the behavior of physical systems. The Tarzan and Jane exercise is rooted in mechanics, as it combines principles of energy conservation with the forces acting on the system. By considering the mechanics of Tarzan's swing, the combined masses, and the pivotal concepts of kinetic and potential energy, one can determine the behavior of the system at different points in time, ultimately leading to the calculation of the final angle of the vine.