An object of mass 5 kg is projected with a velocity of `20 m s^(-1)` at an angle of `60^@` to the horizontal. At the highest point of its path, the projectile explodes and breaks up into two fragments of masses 1 kg and 4 kg. The fragments separate horizontally after the explosion, which releases internal energy such that K.E. of the system at the highest point is doubled Find the separation between the two fragments when they reach the ground.

Answer

5

Explanation

The height of `H = (u^2 sin^2 theta)/(2g) = (20 xx 20)/(2 xx 10) xx 3/4 = 15m`
At the highest point the projectile explodes and breaks into two particles which separate horizontally. As there is no external force acting in horizontal direction, hence linear momentum in horizontal direction should be conserved just before and just after explosion.
Applying conservation of linear momentum
`5 xx 20 cos 60^@ = 4 v_1 - 1v_2 implies 4v_1 - 1 v_2 = 50 " "......(i)`
The kinetic energy at the highest point initially
`K_("initially") = 1/2 xx 5 xx 10 xx 10 = 250 J`
`:.` The kinetic energy after collision
`K_("final") = 2 xx 250 = 500 J`
`implies 1/2 xx 4 xx v_1^2 + 1/2 xx 1 xx v_2^2 = 500`
`implies 4 v_1^2 + v_2^2 = 1000 " ".....(ii)`
After solving (i) and (ii) we get `implies v_1 = 15 m//s "or" 5 m//s and v_2 = + 10m//s or –30 m//s`

Taking the pair `v_1 = 15 m//s and v_2 = +10 m//s`.

The relative velocity of separation of fragments in horizontal direction, `(vret)_x = 25 m//s`
Let the time taken by each fragment to reach the ground be t
Using `H = 1/2 gt^2 , 15 = 1/2 xx 10 xx t^2 implies t = sqrt(3) "sec"`
Hence the separation between the fragments when they reach the ground `x = (vret)_xt`
`implies x = 25sqrt(3)m`.