Question

A proton of mass m collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of 90° with respect to each other. The mass of unknown particle is :
1) 2m 2) m
3) m/2 4) m/√3

Answer 1

Let initially, proton and unknown particle move in x - direction, Let initial velocity of proton is u m/s. after collision, proton makes 45° with x- axis counterclockwise while unknown particle makes 45° with x- axis in clockwise .
so, velocity of proton after collision , $v_1cos45\hat{i}+v_1sin45\hat{j}$
velocity of particle after collision,
$v_2cos45\hat{i}-v_2sin45\hat{j}$

now, from law of conservation of linear momentum,
along x direction,
$mu=mv_1cos45+Mv_2cos45$

$mu=\frac{1}{\sqrt{2}}[mv_1+Mv_2]$....(1)

along y - direction,
$0=mv_1sin45-Mv_2sin45$

$mv_1=Mv_2$.......(2)

coefficient of restitution =$\frac{v_2-v_1cos90}{ucos45}=1$
[for elastic collision, e = 1 ]

so, $\frac{v_2}{ucos45}=1$

$v_2=\frac{u}{\sqrt{2}}$.....(3)

from equations (1), (2) and (3),
we get, M = m

hence, mass of unknown particle is same as mass of proton .

Answer 2

During elastic collision at 90° the mass of unknown particle is m