Center of Mass

Step 1: Find the final velocities after the collision
The first particle has mass 8µg and moves with speed u1. The second particle has mass 4µg and is at rest, so u2=0.
The formula for the final speed of the first particle after an elastic, one-dimensional collision is: v1=‌

(m1−m2)u1+2m2u2

m1+m2

Plug in the numbers:
v1=‌

(8−4)u1+2×4×0

8+4

=‌

4u1

12

=‌

u1

3

The formula for the final speed of the second particle is: v2=‌

2m1u1+(m2−m1)u2

m1+m2

Plug in the numbers:
v2=‌

2×8u1+(4−8)×0

8+4

=‌

16u1

12

=‌

4

3

u1
Step 2: Find the de-Broglie wavelength for each particle
The de-Broglie wavelength formula is: λ=‌

h

mv

where h is Planck's constant, m is mass, and v is velocity.
To compare their wavelengths, find the ratio: ‌

λ2

λ1

=‌

m1v1

m2v2

Plug in the values found earlier: ‌

λ2

λ1

=‌

8× u1 3

4×‌ 4 3 u1

Simplify the expression: =‌

8× u1 3

4×‌ 4u1 3

=‌

8u1∕3

16u1∕3

=‌

8u1

16u1

=‌

1

2

So, ‌

λ1

λ2

=2 which means the ratio of their wavelengths after the collision is 2:1