Question

A uranium nucleus 238U may stay in one piece for billions of years, but sooner or later it decays into a particle of mass 6.64 X 10^27 kg and 234 Th nucleus of mass 3.88 X 10^-25 kg. and the decay process itself is extremely fast (it takes about 10^-2O s). Suppose the uranium nucleus was at rest just before the decay. If the a particle is emitted at a speed of 5.94 X10^6 m/s,

what would be the recoil speed of the thorium nucleus? Answer in units of m/ s

Answer 1

A uranium nucleus 238U may stay in one piece for billions of years, but sooner or later it decays into an alpha particle of mass 6.64 X 10^-27 kg and 234Th nucleus of mass 3.88 X 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 alpha particle is emitted at a speed of 2.55 X 10^7 m/s, what would be the recoil speed of the thorium nucleus? Answer in units of m/s.

All help would be appreciated, since I don't think I remember doing this. We're doing momentum right now and all the problems aren't hard, but I'm stumped on this one.

The 238U nucleus decays (breaks up) and becomes two particles : 234Th and an alpha particle, 4He. (an alpha particle is essentially the nucleus of a Helium atom.)
The numbers in front of the element symbols are the mass numbers. You can see that it balances : 238 = 234 + 4.

You can apply the principle of conservation of momentum to this problem.
Since the uranium nucleus was initially at rest, initial momentum of the system is 0. When the uranium breaks-up, an alpha particle flies off in one direction, and a Thorium nucleus flies off in the opposite direction (recoils). We know that the Thorium nucleus has to fly off in the opposite direction because the algebraic sum of the momentum of the alpha particle and the Thorium nucleus will have to be zero (since initial momentum was zero)
We can write the mathematical equation as :

momentum before decay = momentum after decay
0 = M?V? + M(Th)V(Th)

where M?,V? are the mass and velocity of the alpha particle, and M(Th), V(Th) are the mass and velocity of the Thorium nucleus.

You can find the velocity of the Thorium nucleus because it is the only unknown in the equation.

0 = (6.64 x10^-27)(2.55 x 10^7) + (3.88 x 10^-25)V(Th)
V(Th) = 4.36 x 10^5 m/s

V(Th) is negative because it is opposite to the velocity of the alpha particle.