Projectile Explosion in air

A projectile is fired at an angle of 45° with initial kinetic energy E0. At the top of its trajectory, the projectile explodes with additional energy E0 into two fragments. One fragment of mass m1 travels straight down. What is the velocity (magnitude and direction) of the second fragment of mass m2 and the velocity of the first? What is the ratio of m1/m2 when m1 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 m1 and m2 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 vy and v1 gives

m1 travels straight down with speed = v1
m2 travels in the y-z plane

The mass m1 is the largest it can be when v1 = 0 , meaning 2m2 = m1 and the mass ratio is