A uranium nucleus 238U may stay in one piece for billions of years, but sooner or later it decays into an a particle of mass 6.64×10^−27 kg and 234Th nucleus of mass 3.88 × 10^−25 kg, and the decay process itself is extremely fast(it takes about 10^−20 s). Suppose the uranium nucleus was at rest just before the decay. If the a particle is emitted at a speed of 1.07×10^7 m/s, what would be the recoil speed of the thorium nucleus?

Question

A uranium nucleus 238U may stay in one piece for billions of years, but sooner or later it decays into an a particle of mass 6.64×10^−27 kg and 234Th nucleus of mass 3.88 × 10^−25 kg, and the decay process itself is extremely fast(it takes about 10^−20 s). Suppose the uranium nucleus was at rest just before the decay. If the a particle is emitted at a speed of 1.07×10^7 m/s, what would be the recoil speed of the thorium nucleus?

Answer in units of m/s.

I got -183113.4021m/s, but that was wrong, and so was 183113.4021m/s

Answer 1

Don't use so many significant figures.

I get 1.83*10^5 m/s

Answer 2

To find the recoil speed of the thorium nucleus, we can use the principle of conservation of momentum. Before the decay, the uranium nucleus is at rest, so its momentum is zero. After the decay, the total momentum of the system must remain zero.

The momentum equation can be written as:
Initial momentum = Final momentum

Since the uranium nucleus is at rest initially, the momentum of the alpha particle and the thorium nucleus must cancel each other out. Let us denote the recoil speed of the thorium nucleus as v.

Initial momentum: 0 (since uranium nucleus is at rest)
Final momentum: momentum of the alpha particle + momentum of the thorium nucleus

The momentum of an object is given by the product of its mass and velocity. Let's calculate the momentum of the alpha particle and the thorium nucleus:

Momentum of the alpha particle = mass of the alpha particle * velocity of the alpha particle
Momentum of the thorium nucleus = mass of the thorium nucleus * velocity of the thorium nucleus

The mass of the alpha particle is 6.64×10^−27 kg, and its velocity is given as 1.07×10^7 m/s. The mass of the thorium nucleus is 3.88 × 10^−25 kg.

Therefore, we can rewrite the momentum equation as:

0 = (mass of the alpha particle * velocity of the alpha particle) + (mass of the thorium nucleus * v)

Solving for v, we get:

v = -(mass of the alpha particle / mass of the thorium nucleus) * (velocity of the alpha particle)

Substituting the values, we get:

v = - (6.64×10^−27 kg / 3.88 × 10^−25 kg) * (1.07×10^7 m/s)

Calculating the value, we get:

v = - 0.1729 * 1.07×10^7 m/s

v ≈ -1.85 × 10^6 m/s

Therefore, the recoil speed of the thorium nucleus is approximately -1.85 × 10^6 m/s. The negative sign indicates that the direction of the recoil is opposite to the direction of the alpha particle.