A nucleus disintegrates into two nuclear parts, in such a way that ratio of their nuclear sizes is 1: 2^(1 / 3). Their respective speed have a ratio of n: 1. The value of n is __________.

Solution

Physics sir · 3 steps

Turn the size ratio into a mass ratio first, then let momentum conservation give the speeds.

1. Size to mass. Nuclear radius goes as R = R₀ A^(1/3), so A ∝ R³. Cubing 1 : 2^(1/3) gives the mass ratio A₁ : A₂ = 1 : 2.
2. Momentum is conserved. The nucleus starts at rest, so the two fragments fly apart with equal and opposite momenta.

   m₁ v₁ = m₂ v₂ ⇒ v₁/v₂ = m₂/m₁ = 2/1

3. So. The smaller fragment moves twice as fast, so v₁ : v₂ = 2 : 1 and n = 2.

The common slip: 0.5. Taking speed proportional to mass instead of inversely proportional gives v₁ : v₂ = 1 : 2, so n = 0.5. The lighter fragment is the faster one.

The idea to keep. Nuclear radius goes as A^(1/3), so cube a size ratio to get the mass ratio. When a nucleus at rest breaks in two, the speeds are in the inverse ratio of the masses.

Answer from the NTA final answer key. Solution by PrepWiser.