What fraction of its mechanical energy is lost in each bounce, if a ball bounces to 80% of its

original height?

When a ball bounces to 80% of its original height, it’s a great example of how energy is conserved and transformed in physics. Let's break down how we determine the fraction of mechanical energy lost during each bounce.

Understanding Energy Transformation

When the ball is dropped, it possesses gravitational potential energy, which is converted to kinetic energy as it falls. Upon hitting the ground, some of this energy is transformed back into potential energy as the ball bounces back up. However, because the ball only reaches 80% of its original height, we know that some energy is lost, likely due to factors like air resistance and inelastic deformation of the ball.

Calculating the Original and Rebound Heights

Let’s denote the original height from which the ball is dropped as h. The potential energy at this height can be expressed as:

• PE_initial = mgh

where m is the mass of the ball, g is the acceleration due to gravity, and h is the height. After the bounce, the ball reaches 80% of this height, which is 0.8h. The potential energy at this new height becomes:

• PE_rebound = mg(0.8h) = 0.8mgh

Finding the Energy Lost

To find the energy lost during the bounce, we can subtract the potential energy after the bounce from the potential energy before the bounce:

• Energy Lost = PE_initial - PE_rebound
• Energy Lost = mgh - 0.8mgh
• Energy Lost = mgh(1 - 0.8) = mgh(0.2)

Determining the Fraction of Energy Lost

To find the fraction of mechanical energy lost, we take the energy lost and divide it by the initial potential energy:

• Fraction Lost = (Energy Lost) / (PE_initial)
• Fraction Lost = (0.2mgh) / (mgh)
• Fraction Lost = 0.2

Final Thoughts

This calculation shows that the ball loses 20% of its mechanical energy in each bounce, which is consistent with the fact that it only reaches 80% of its original height. Understanding these concepts not only highlights the energy transformations at play but also gives insight into the efficiency of different materials and conditions affecting the bounces of balls in real-world scenarios.