## Projectile Explosion in air

A projectile is fired at an angle of 45° with initial kinetic energy E_{0}. At the top of its trajectory, the projectile explodes with additional energy E_{0} into two fragments. One fragment of mass m_{1} travels straight down. What is the velocity (magnitude and direction) of the second fragment of mass m_{2} and the velocity of the first? What is the ratio of m_{1}/m_{2} when m_{1} is a maximum?

Solution :

Let the axes be as shown with the projectile in the y-z plane. At the top just before the explosion, the velocity is in the y direction and has magnitude

where m_{1} and m_{2} are the masses of the fragments. The initial momentum is

The final momentum is

The conservation of momentum equations are

The energy equation is

Substituting for v_{y} and v_{1} gives

m_{1} travels straight down with speed = v_{1}

m_{2} travels in the y-z plane

The mass m1 is the largest it can be when v_{1} = 0 , meaning 2m_{2} = m_{1} and the mass ratio is