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 1.07×107 m/s, what would be the recoil speed of the thorium nucleus?
Answer in units of m/s.
Conservation of momentum requires that:
(alpha particle mass)x(alpha particle speed) = (Th234 speed)x (Th234 mass)
The speeds are in opposite directions
Therefore
(Th34 speed)= (alpha speed)*(4/234)
To find the recoil speed of the thorium nucleus (234Th), we can apply the principle of conservation of momentum. The total momentum before the decay must be equal to the total momentum after the decay, assuming no external forces are acting on the system.
Before the decay:
The uranium nucleus (238U) is at rest, so its initial momentum is zero (0 kg m/s).
After the decay:
The alpha particle (a) is emitted with a mass of 6.64×10−27 kg and a speed of 1.07×10^7 m/s. The thorium nucleus (234Th) is initially at rest.
Let's assume the recoil speed of the thorium nucleus after the emission of the alpha particle is v (in m/s).
Applying the conservation of momentum, we have:
(Initial momentum) = (Final momentum)
0 = (Mass of alpha particle) x (Speed of alpha particle) + (Mass of thorium nucleus) x (Recoil speed of thorium nucleus)
0 = (6.64×10−27 kg) x (1.07×10^7 m/s) + (3.88×10−25 kg) x (Recoil speed)
Simplifying the equation and solving for the recoil speed:
Recoil speed = - (6.64×10−27 kg) x (1.07×10^7 m/s) / (3.88×10−25 kg)
Recoil speed = -1.816 m/s
The negative sign indicates that the recoil speed of the thorium nucleus is in the opposite direction to that of the alpha particle.