Question

Aldo and his brother Pablo have gone to skate to a frozen lagoon. At one point he stops to open his lunch, when he observes that Pablo is approaching directly towards him with a speed V0. Before the impact, Aldo throws his brother the lunch package in the horizontal direction which is caught by Pablo in the air. If both brothers have the same mass M and the mass of the package is m, calculate the minimum value of how quickly Aldo should throw the package so that his brother Pablo does not clash with him.

Answer 1

For Pablo to not clash with Aldo, the velocity of Pablo with respect to Aldo should be zero.

If Aldo throws the package horizontally towards Pablo with speed , the horizontal velocity of the package remains unchanged upto the time it is caught by Pablo because there is no force on the package in the horizontal direction until it is caught by Pablo.

After the package is caught by Pablo, the total mass of Pablo and the package is   .

Assuming Pablo is just gliding towards Aldo, i.e. Pablo is not applying any force to change his speed and there is no friction, we can use the fact that the total horizontal momentum of Pablo and the package should be conserved because there is no horizontal force on Pablo+Package system after the package has been thrown by Aldo.

The total horizontal momentum of Pablo and the  package before it is caught by Pablo is

Note that momenta of Pablo and the package, before it is caught by Pablo, have opposite direction because they are moving in opposite directions.  

If is at least such that above total horizontal momentum is zero, then after catching the package Pablo will stop moving towards Aldo because we now have a combined system of mass whose momentum will be zero only if its velocity is zero.

So, we find that   is the minimum horizontal speed with which Aldo should throw the package towards Pablo. If Aldo throws the package with a horizontal speed greater than , the conservation of momentum implies that the combined system of mass , after Pablo catches the package, moves away from Aldo with speed such that

So,