To investigate the relationship between the height a ball is dropped from and the height of its rebound.
When a ball is travelling to and from the ground, some of its energy is lost due to the work done by frictional forces — air resistance. These effects are taken to be negligible in this explanation of the physics underlying this investigation.
When a ball is held at a height above the ground, it has gravitational potential energy — $E_P$. The magnitude of its $E_P$ is determined by the equation $E_P=mgh$, where $m$ is the ball’s mass — measured in $kg$, $g$ is the gravitational field strength acting on the ball — measured in $Nkg^{-1}$, and $h$ is the height — measured in $m$.
As the ball falls it looses $E_P$ and gains kinetic energy — $E_K$, i.e. the energy is converted from one form to the other.
Upon hitting the ground the ball’s kinetic energy begins converting into elastic potential energy — $E_{EP}$ — as the ball is deformed, however some energy is also lost to the surroundings as it is converted sound and heat energy upon impact, meaning that $E_{EP}<E_K$ (its kinetic energy when reaching the ground). Because some of the initial $E_P$ is always lost, this means a ball’s rebound height is always less than its initial drop height.
Once the ball reaches the limit for the elastic potential energy it can store it begins to regain its original shape and ‘rebounds’ upwards — $E_{EP}$ is converted back to $E_K$. As the ball rises its $E_K$is converted into $E_P$ again, and the process repeats itself, until all the ball’s energy has been dissipated.
The ball’s rebound height is given by $h={E_P \over mg}$, where $E_P$ was the potential energy it had upon its rebound, which is, in effect, equal to its $E_K$ when it leaves the ground. Therefore the factors that significantly impact the rebound height of a ball are those which impact its $E_K$ when it leaves the ground.
There are many factors this could be, such as elasticity, ground surface material, size, etc. In this investigation, all those factors are attempted to be controlled and the one in question is drop height.
From the equation ${E_P \over h}=mg$, it is easy to see that the height of the ball and the gravitational potential energy stored in it have a positive correlation, i.e. the greater $h$ the greater $E_P$, for a constant $m$ and $g$. A greater initial $E_P$ of the ball means that, its $E_K$ upon leaving the ground will be greater — because, although some of this energy will be lost when it hits the ground in the form of sound and heat energy, if all other factors are controlled (i.e. the same magnitude of energy is lost for an increasing initial $E_P$) the ball will have more energy upon rebounding. This greater energy means it can rise farther (i.e. its final $E_P$ will be greater) which will lead to a greater rebound height.
$\therefore$ For a ball with a greater drop height, a greater rebound height should be observed. This relationship is, in fact, recorded in the experimental investigation.