How is kinetic energy mathematically expressed?

Kinetic energy is mathematically expressed as ( KE = \frac{1}{2} mv^2 ). This equation defines kinetic energy as being proportional to the mass of an object (m) and the square of its velocity (v). The reasoning behind this formula is grounded in physics, where kinetic energy represents the energy an object possesses due to its motion.

The factor ( \frac{1}{2} ) is necessary because the kinetic energy increases with the square of the velocity; as an object's speed doubles, its kinetic energy increases fourfold, indicating a quadratic relationship. This relationship is fundamental in both classical mechanics and various applications involving motion.

Other forms presented in the choices do not represent kinetic energy. The formula ( KE = mv ) describes momentum, while ( KE = mgh ) is the equation for gravitational potential energy, indicating energy stored due to an object's position in a gravitational field. The expression ( KE = p/t ) does not correspond to any known energy equation in classical mechanics. Thus, the correct mathematical representation of kinetic energy is ( KE = \frac{1}{2} mv^2 ).