COMEDK 2023 Shift 1 Paper

The de Broglie wavelength of a particle is given by the formula:
λ=‌

h

p

where h is Planck's constant and p is the momentum of the particle. The momentum p can be expressed in terms of the mass m and velocity v of the particle as p=m⋅v.
For the two particles in question, their de Broglie wavelengths are given by:
‌λ1=‌

h

p1

=‌

h

m1v1

‌λ2=‌

h

p2

=‌

h

m2v2

To find the ratio ‌

λ1

λ2

, we divide the equations:
‌

λ1

λ2

=‌

‌ h m1v1

h m2v2

=‌

m2v2

m1v1

‌. ‌
Given that the original particle is at rest before decaying into two particles, by the conservation of momentum, the total momentum before decay must equal the total momentum after decay. Before the decay, the momentum is 0 (since the particle is at rest), and after the decay, it must still sum to 0 . Thus, we have:
m1v1=m2v2
Substituting m1v1=m2v2 into the equation for the ratio of the de Broglie wavelengths, we get:
‌

λ1

λ2

=‌

m2v2

m1v1

=1
Therefore, the ratio of their de Broglie wavelengths ‌

λ1

λ2

is 1:1. The correct option is:
Option B 1 : 1