How does the height from which an object falls affect its final velocity?

Answers:

• The height from which an object falls affects its final velocity due to the principles of energy conservation and the equations of motion under the influence of gravity. When an object falls freely under the influence of gravity, it converts potential energy into kinetic energy.

  1. Potential Energy and Kinetic Energy: When an object is at a height ( h ) above the ground, it has gravitational potential energy given by the formula: [ PE = mgh ] where ( m ) is the mass of the object, ( g ) is the acceleration due to gravity (approximately ( 9.81 , \text{m/s}^2 ) near the Earth's surface), and ( h ) is the height.

     As the object falls, this potential energy is converted into kinetic energy, which is given by: [ KE = \frac{1}{2}mv^2 ] where ( v ) is the final velocity of the object just before impact.

  2. Conservation of Energy: In the absence of air resistance, the total mechanical energy remains constant. Therefore, the potential energy at the height ( h ) will equal the kinetic energy right before hitting the ground: [ mgh = \frac{1}{2}mv^2 ] The mass ( m ) can be cancelled out from both sides: [ gh = \frac{1}{2}v^2 ]

  3. Solving for Final Velocity: Rearranging the equation gives: [ v^2 = 2gh ] Taking the square root of both sides yields the final velocity just before impact: [ v = \sqrt{2gh} ]

  This equation shows that the final velocity ( v ) of the object just before it hits the ground is directly related to the square root of the height ( h ) from which it falls. Therefore, if you increase the height ( h ), the final velocity ( v ) increases as well.

  Conclusion:

  In summary, the height from which an object falls affects its final velocity according to the relationship: [ v = \sqrt{2gh} ] This means that a larger drop height results in a larger final impact velocity, assuming negligible air resistance and other forms of energy dissipation.