Center of Mass

Given:
The final velocity of the first sphere, v1=‌

7

9

u1
u2=0 (the second sphere is at rest)
The mass of a sphere is proportional to the cube of its radius: m∝R3
Let m1 and m2 be the masses, and R1 and R2 be the radii of the first and second spheres.
Step 1: Use the formula for an elastic collision
After the collision, the velocity of the first sphere is:
v1=‌

(m1−m2)u1+2m2u2

m1+m2

Step 2: Substitute u2=0
v1=‌

(m1−m2)u1

m1+m2

We know v1=‌

7

9

u1 :
‌

7

9

u1=‌

(m1−m2)u1

m1+m2

Divide both sides by u1 (since u1≠0 ):
‌

7

9

=‌

m1−m2

m1+m2

Step 3: Solve for m1 in terms of m2
Multiply both sides by ( m1+m2 ):
‌7(m1+m2)=9(m1−m2)
‌7m1+7m2=9m1−9m2
‌7m2+9m2=9m1−7m1
‌16m2=2m1
‌m1=8m2
Step 4: Relate masses to radii
Since mass of a sphere, m∝R3 :
‌‌

m1

m2

=(‌

R1

R2

)3
‌8=(‌

R1

R2

)3
Step 5: Solve for R2
Take the cube root of both sides:
‌‌

R1

R2

=2
‌R1=2R2

R2=‌

R1

2

Plug in R1=1.2cm :
R2=‌

1.2

2

=0.6cm