A basketball is released from a height of 3m onto a hard surface. What height will the ball rebound to and why ?
To determine the height the ball rebounds to, we need to consider the principle of conservation of energy. When the ball hits the hard surface, it undergoes an elastic collision, which means kinetic energy is conserved. However, potential energy is converted into kinetic energy during the collision.
The potential energy (PE) of the ball at its initial height is given by the equation PE = mgh, where m is the mass of the ball (assumed to be constant), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (3m).
When the ball rebounds, it reaches a certain height, h'. At that new height, the potential energy is converted back into kinetic energy and then into potential energy once again as it reaches its maximum height.
Since the ball is in contact with the hard surface for a very short time, we can assume the gravitational force has negligible effect during this time. Therefore, the only energy transfer is between potential and kinetic.
Considering the conservation of energy, the initial potential energy (PE) is equal to the final potential energy (PE') plus the final kinetic energy (KE'). Mathematically, this can be written as:
PE = PE' + KE'
Substituting the expressions for potential energy, we have:
mgh = mgh' + KE'
Since the ball rebounds elastically, meaning the kinetic energy is not lost, we can rewrite the equation as:
mgh = mgh' + (1/2)mv^2
Where v is the velocity of the ball just before impact with the hard surface.
Now, let's solve for h', the rebound height:
h' = (v^2 - gh) / (2g)
To find v, we can apply another conservation principle, conservation of mechanical energy. Since the ball falls freely before hitting the surface, its initial kinetic energy (KE) is equal to its final potential energy (PE'):
KE = PE'
(1/2)mv^2 = mgh'
Simplifying and solving for v, we get:
v = sqrt(2gh')
Substituting this value back into the equation for h', we can determine the rebound height. However, without information about the coefficient of restitution (a measure of how "bouncy" the ball is), we cannot give a specific numerical answer.
In conclusion, the height the ball rebounds to depends on its initial release height, the coefficient of restitution, and the velocity just before impact with the hard surface.