Two particles, A and B of masses m and 3m, are moving along X and Y axes respectively, with the same speed v. They collide at the origin, and coalesce into one body, after the c ollision. What is the velocity of this coalesced mass ?

Explanation

Let V be the speed of the combined mass and let `vecV` make an angle `alpha` with + ve x-axis
`mv=4mV c osalpha" "…(i)` (conservation of momentum of x component)
`3mv =4mv sin alpha" "…(ii)`
Find `tanalpha,`
Find `cos alpha and sin alpha` from (i) & (ii) and put in `cos^(2)alpha+sin^(2)alpha=1`
This will give V in terms of v .

Figure shows the two particles before and after the collision. Let V be the speed of the combined mass and let the direction of `vecV` be making an angle `alpha` with the positive X-axis, after collision Use law of conservation of linear momentum
`mv=4mVcosalpha" "...(i)` (For X-components)
`3mv=4mV sin alpha" "...(ii)` (For Y-components)
Divide equation (ii) by equation (i)
`(4mV sinalpha)/(4mV cosalpha)=(3mv)/(mv)`
`tan alpha=3`
`{:(alpha=tan^(-1)(3),...(iii)),("From"(i)cosalpha=(v)/(4V),...(iv)),("From"(ii)sinalpha=(3v)/(4V),...(v)),(cos^(2)alpha+sin^(2)alpha=1,...(iv)):}`
Put values from (iv) & (v) in (vi)
`((v)/(4V))^(2)+((3v)/(4V))^(2)=1`
`v^(2)((v)/(4))^(2)+((3v)/(4))^(2)=10/16v^(2)=5/8v^(2)`
`V=sqrt((5)/(8))v`