Irodov Solution 1.158

The first time the ball hits the surface, it has fallen a height h i.e. lost a potential energy of mgh. This potential energy is converted into its kinetic energy and so if it hit the surface with a speed v0, then,





After hitting the surface, its upwards velocity will reduce to . So when the ball returns to the surface to bounce a second time it impacts the surface with a speed . This time when the ball rises up it will loose a fraction of its impact velocity and bounce up with a speed . This process will continue with each bounce. In other words the ball will loose its impact velocity in a geometric progression with each bounce following the sequence .
At the kth bounce the ball impacts the surface at a speed and then reflects back with a speed . During this bounce the momentum of the ball changes from to (the minus sign comes because the direction of velocity has reversed to up after the bounce, we have chosen the positive direction as downwards). This change is momentum is what the surface absorbs from the impact of the ball. So at the kth bounce, the ball imparts a momentum of




The total momentum imparted by the ball it the sum of the momenta imparted after each bounce and is given by the summation of the geometric series,

Irodov Problem 1.157

Let the mass per unit length of the falling chain be . The mass of an infinitesimally piece of chain of length dl is then given by. The momentum imparted by a piece of chain that is dl long and that hits the table with a speed v is given by
.





If this infinitesimally small piece of chain of length dl falls in time dt, the instantaneous force imparted by it (force is change in momentum per time) on the table at a speed v is given by,
. After a time t has elapsed since the chain starts to fall, the velocity of the tip of the chain that is hitting the table will be v=gt since it has been falling under the influence of gravity. This means that the force exerted by the falling chain as a function of time t will be .



In time t since the chain starts to fall, a length of would have fallen on the table (since it is freely falling under gravity). This means that after a time t has elapsed since the chain starts to fall the mass that lies on the table would be . The force exerted by this part of the chain would then be . Clearly this is half of the force exetrted by the falling chain.