Question

A model rocket is fired from the ground in a parabolic arc. At the very top of the arc, at a horizontal displacement of 260 m from the launch point, an explosion occurs within the rocket, breaking it into two fragments which separate from each other. One fragment, having 1/3 the mass of the rocket, falls straight down to Earth as if it were at rest at that moment right after the explosion.  Determine the total horizontal displacement from the launch point the other fragment lands.  (The two fragments land simultaneously.)

Answer 1

At the top of the arc there will be only hrizontal velocity before explosion and after explosion on fragment which has 1/3 mass will fall horizontally from the height as if it was at rest and other part which has 2/3 mass of original rocket

let velocity of original rocket before explosion be u and it's mass be m and let after explosion velocity of heavier part that has 2/3 mass of original one have v velocity and other part have velocity eqaul to zero

applying conservation of linear momentum before and after explsion

mu=2mv/3

v=3u/2 along the horizontal direction

since both fragment takes same time to reach to the ground and heavier part has velocity along horizontal direction hence the heavier part will follow the horizontal projecticle with initial horizontal velocity 3/2 times the velocity of original rocket hence the fragment will travese another 3/2 times the horizontal range covered by rocket before explosion.

Hence total horizontal range from begining to the final landing of the fragments= 260+ 3*260/2 = 650 m