91Ó°ÊÓ

Kinetic energy is the energy that an object possesses due to its motion. Picture a ball being thrown upward; as it moves, it carries kinetic energy. This can be calculated with the formula: \[ KE = \frac{1}{2}mv^2 \]Here, \( m \) represents the mass of the object, and \( v \) represents its velocity. In our case, the ball has a mass of \( 0.63 \, \text{kg} \) and an initial speed of \( 14 \, \text{m/s} \).

• Kinetic energy is significant when an object is in motion.
• The faster the object moves, the higher its kinetic energy.
• For the ball, its kinetic energy when thrown is \[ KE_{\text{initial}} = \frac{1}{2} \times 0.63 \, \text{kg} \times (14 \, \text{m/s})^2 \]

As the ball ascends and slows down, its kinetic energy decreases, converting to potential energy.

Potential energy is the stored energy of an object based on its position or height. When the ball reaches its maximum height, its kinetic energy becomes negligible and it has maximum potential energy. The potential energy has increased as the ball moved against the gravitational force.

• Potential energy depends on the height above the ground and gravitational acceleration.
• The formula to compute gravitational potential energy is \[ PE = mgh \] where \( m \) is mass, \( g \) is the gravitational acceleration (approximately \( 9.8 \, \text{m/s}^2 \) on Earth), and \( h \) is the height.

In this scenario, when the ball is at its maximum height of \( 8.1 \, \text{m} \), its potential energy will be:\[ PE_{\text{final}} = 0.63 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 8.1 \, \text{m} \]This represents the energy stored due to the ball's raised position.

The principle of conservation of energy states that the total energy in an isolated system remains constant. As the ball is thrown upward, the system transitions energy between kinetic and potential forms but the total mechanical energy is conserved.

• Initially, the ball has high kinetic energy and low potential energy since it’s just starting its ascent from the ground.
• As the ball reaches the top of its path, most kinetic energy has been converted into potential energy.

Mechanical energy is the sum of kinetic and potential energy. Even though they shift between each other, their total remains unchanged during the motion when air resistance and other external forces are negligible:\[ E_{\text{mechanical}} = KE + PE \]In this way, there is no net change in the mechanical energy of the ball-Earth system during its ascent, assuming no external work is done. During the ascent to its maximum height, the kinetic energy initially decreases, converting into potential energy, but the total sum remains the same.