Question

An object thrown in the air has a velocity after t seconds described by v(t) = -9.8t + 31 (in meters/second) and a height h(t) = -4.9t^2 + 31t + 45 (in meters). The object has mass m = 4 kilograms. The kinetic energy of the object is given by K = 1/2 mv^2, and the potential energy is given by U = 9.8mh. Enter an expression for the total kinetic and potential energy K + U as a function of time. What does this expression tell you about the energy of the falling object? Complete the explanation.

K + U = (____) joules

Since the sum is (U/K/a single variable/a constant), this means that the energy of the object is constant, and as it (gains/loses) kinetic energy by falling, it (gains/loses) the same amount of potential energy.

Answer 1

K + U = 1/2 * mv² + 9.8 * mh

Step-by-step explanation:

The total energy (K + U) of the falling object is the sum of its kinetic energy (K) and potential energy (U). The expressions for kinetic and potential energy are given by:

K = 1/2 * mv²

U = 9.8 * mh

Substituting these into the total energy expression:

K + U = 1/2 * mv² + 9.8 * mh

This expression represents the total mechanical energy of the object at any given time during its fall. The term 1/2 * mv² is the kinetic energy, which depends on the velocity of the object, and 9.8 * mh is the potential energy, which depends on the height of the object and the acceleration due to gravity (9.8 m/s²).

The sum K + U is constant throughout the fall, as no external forces (like air resistance) are considered. This is a result of the conservation of mechanical energy, stating that the total mechanical energy of a system remains constant if only conservative forces (like gravity) are at play. As the object falls, it converts potential energy into kinetic energy and vice versa, but the total remains constant.

During the fall, the object gains kinetic energy as it accelerates downward and loses potential energy as it descends. This is reflected in the equation K + U, where the gain in kinetic energy is accompanied by a proportional loss in potential energy.