91Ó°ÊÓ

In physics, kinetic energy represents the energy a particle possesses due to its motion. The formula to calculate kinetic energy is \( KE = \frac{1}{2} mv^2 \) , where \( m \) denotes mass and \( v \) is the velocity of the particle. This energy depends on both the mass and speed, showing how much energy an object has when it moves.
Consider a particle projected with an initial kinetic energy \( E \) at an angle to the horizontal. Its initial velocity is divided into horizontal and vertical components. At the top of its trajectory, only the horizontal component contributes to the kinetic energy.

• The horizontal component of the velocity remains constant throughout the projectile's flight.
• Vertical component becomes zero at the highest point of the trajectory.

Thus, the kinetic energy at the highest point is derived solely from the horizontal motion, calculated using \( \frac{1}{4} mv^2 \) , implying half the initial kinetic energy, owing to the squared terms in the wave

Potential energy is the stored energy of position possessed by an object. For a particle in projectile motion, the potential energy is related to its height above the ground. We calculate it using the formula \( PE = mgh \) , where \( g \) stands for the acceleration due to gravity, and \( h \) is the height.
When the particle reaches its highest point, it has gained the most height. At this peak, the potential energy captures the initial energy that is not used for horizontal kinetic energy. For our projectile, given the total energy initially is \( E \) and has undergone horizontal kinetic energy, its potential energy at the apex is \( E - \frac{E}{2} = \frac{E}{2} \) . This demonstrates energy transformation from kinetic to potential as the projectile rises.

• Potential energy rises as height increases.
• At maximum height, the potential energy is greatest in projectile motion.

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In projectile motion, the total mechanical energy (kinetic plus potential) remains constant, provided there is no air resistance.
Initially, when the particle is launched, it has maximum kinetic energy and minimum potential energy. As it rises, the kinetic energy decreases while potential energy increases, maintaining the total energy \( E \).

• At launch, high kinetic energy, low potential energy.
• At highest point, energy is equally divided between kinetic and potential forms.
• During descent, potential energy converts back to kinetic.

By understanding energy conservation, we see how the total \( E \) stays the same, providing insights into energy's behavior in projectile motion. It clearly illustrates the interchange between kinetic and potential energies throughout the trajectory.