a uranium-238 nucleus may stay in one piece for billions of years, but later it decays into an a particle of mass 6.64 x 10^-27 and a thorium-234 nucleus of mass 3.88 x 10^-25. The decay process itself is extremely fast (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 1.4 x 10^7 m/s, what would be the recoil speed of the thorium nucleus?
Please help. I don't know where to begin.
Use the law of conservation of mometum. the total momentum of the two particles before the ejection was zero...
0=mass234*Vth+ massalpha*Valpha
solve for Valpha
To find the recoil speed of the thorium nucleus, we can use the principle of conservation of momentum. According to this principle, the total momentum before the decay should be equal to the total momentum after the decay.
Let's denote the mass of the uranium nucleus as Mu, the mass of the alpha particle as Ma, and the mass of the thorium nucleus as Mt. The initial velocity of the uranium nucleus is 0 (since it is at rest), and the velocity of the alpha particle is given as 1.4 x 10^7 m/s. We want to find the recoil velocity of the thorium nucleus.
The momentum of an object is given by the product of its mass and velocity. Therefore, the initial momentum before the decay is zero, and the final momentum after the decay should also be zero.
Using the conservation of momentum, we can write the equation as:
0 + (Mu * 0) = (Ma * 1.4 x 10^7) + (Mt * Vt)
Since the initial momentum is zero, the momentum of the alpha particle and the thorium nucleus should cancel each other out.
Now, we can substitute the values given in the question. The mass of the alpha particle, Ma, is 6.64 x 10^-27 kg, and the mass of the thorium nucleus, Mt, is 3.88 x 10^-25 kg. The velocity of the alpha particle, Va, is 1.4 x 10^7 m/s.
0 = (6.64 x 10^-27) * (1.4 x 10^7) + (3.88 x 10^-25) * Vt
Now, we can solve for Vt, the recoil velocity of the thorium nucleus.
Vt = - [(6.64 x 10^-27) * (1.4 x 10^7)] / (3.88 x 10^-25)
Calculating this expression on a calculator will give you the recoil velocity of the thorium nucleus.
Make sure to convert the mass of the thorium nucleus and the recoil velocity to the same units before performing the calculation, if necessary.