Final answer:
The kinetic energy of a block sliding down a frictionless track is calculated using the conservation of energy principle. It is equal to the initial potential energy, which depends on the block's mass and the height from which it slides down, resulting in the equation KE = mgh.
Step-by-step explanation:
To calculate the kinetic energy of a block sliding down a curved, frictionless track, we can apply the principle of conservation of energy. The kinetic energy (KE) of the block at any point on the track is equal to the initial potential energy (PE) at the starting height minus the potential energy at the current height. The formula for kinetic energy is KE = ½mv², where 'm' is the mass of the block and 'v' is its velocity. The potential energy is given by PE = mgh, where 'g' is the acceleration due to gravity (9.8 m/s²) and 'h' is the height.
When the block slides down from height 'h' on a frictionless surface, its potential energy is converted into kinetic energy. As there's no friction to dissipate energy as heat, the total mechanical energy remains constant. Therefore, the kinetic energy of the block at any point equals the initial gravitational potential energy it had at height 'h'. This can be expressed as KE = mgh. The mass of the block and the height from which it slides down are the main factors influencing its kinetic energy.