A single uranium atom has a mass of 3.97 x 10^-25 kg. It decays into the nucleus of a thorium atom by emitting an alpha particle at a speed of 2.10 x 10^7 m/s. The mass of an alpha particle is 6.68 x 10^-27 kg. What is the recoil speed of the thorium nucleus?
Why did the thorium nucleus need a recoil speed? Because it wanted to be the life of the party! But in all seriousness, let's calculate that for you.
To find the recoil speed of the thorium nucleus, we need to apply the law of conservation of momentum. The total momentum before the decay should be equal to the total momentum after the decay.
The momentum of the uranium atom before decay is given by the product of its mass and velocity:
Momentum of uranium = (Mass of uranium) * (Velocity of uranium) = (3.97 x 10^-25 kg) * (2.10 x 10^7 m/s)
The momentum of the alpha particle after the decay is given by:
Momentum of alpha particle = (Mass of alpha particle) * (Velocity of alpha particle) = (6.68 x 10^-27 kg) * (unknown velocity of thorium nucleus)
Since momentum is conserved, we can set these two values equal to each other:
(Mass of uranium) * (Velocity of uranium) = (Mass of alpha particle) * (Velocity of thorium nucleus)
Now we can solve for the velocity of the thorium nucleus:
(3.97 x 10^-25 kg) * (2.10 x 10^7 m/s) = (6.68 x 10^-27 kg) * (Velocity of thorium nucleus)
Rearranging the equation, we find:
Velocity of thorium nucleus = [(3.97 x 10^-25 kg) * (2.10 x 10^7 m/s)] / (6.68 x 10^-27 kg)
Calculating the equation will give us the recoil speed of the thorium nucleus.
To find the recoil speed of the thorium nucleus, we can use the principle of conservation of momentum. The total momentum before the decay is equal to the total momentum after the decay.
Let's denote the recoil speed of the thorium nucleus as V.
Before the decay:
The momentum of the uranium atom is given by:
Momentum1 = mass of uranium atom * speed of uranium atom
Momentum1 = (3.97 x 10^-25 kg) * (2.10 x 10^7 m/s)
The momentum of the alpha particle is given by:
Momentum2 = mass of alpha particle * speed of alpha particle
Momentum2 = (6.68 x 10^-27 kg) * (2.10 x 10^7 m/s)
Total momentum before the decay = Momentum1 + Momentum2
After the decay:
The momentum of the thorium nucleus is given by:
Momentum3 = mass of thorium nucleus * recoil speed of thorium nucleus
Momentum3 = (mass of thorium nucleus) * V
Total momentum after the decay = Momentum3
Now, equating the total momentum before and after the decay:
Momentum1 + Momentum2 = Momentum3
(3.97 x 10^-25 kg) * (2.10 x 10^7 m/s) + (6.68 x 10^-27 kg) * (2.10 x 10^7 m/s) = (mass of thorium nucleus) * V
Solve the equation to find V:
V = [(3.97 x 10^-25 kg) * (2.10 x 10^7 m/s) + (6.68 x 10^-27 kg) * (2.10 x 10^7 m/s)] / (mass of thorium nucleus)
Note: The formula assumes that no other forces are acting on the system, such as external forces.
To find the recoil speed of the thorium nucleus, we can apply the conservation of linear momentum. According to this principle, the total momentum before the decay must equal the total momentum after the decay.
The initial momentum is given by the product of the mass and speed of the uranium atom:
Initial momentum = mass of uranium atom x speed of uranium atom
= (3.97 x 10^-25 kg) x (2.10 x 10^7 m/s)
= 8.337 x 10^-18 kg·m/s
The final momentum is the sum of the momentum of the emitted alpha particle and the recoil momentum of the thorium nucleus. Let's denote the recoil speed of the thorium nucleus as V:
Final momentum = momentum of alpha particle + recoil momentum of thorium nucleus
(mass of alpha particle)(speed of alpha particle) + (mass of thorium nucleus)(recoil speed of thorium nucleus)
= (6.68 x 10^-27 kg)(2.10 x 10^7 m/s) + (unknown)(V)
Since the thorium nucleus is much more massive than the alpha particle, we can ignore the small contribution of the alpha particle's mass.
Hence, the equation becomes:
Final momentum ≈ (mass of thorium nucleus)(recoil speed of thorium nucleus)
Therefore:
(mass of thorium nucleus)(recoil speed of thorium nucleus) ≈ (6.68 x 10^-27 kg)(2.10 x 10^7 m/s)
Now we can solve for the recoil speed of the thorium nucleus:
recoil speed of thorium nucleus ≈ (6.68 x 10^-27 kg)(2.10 x 10^7 m/s) / (mass of thorium nucleus)
The mass of a thorium nucleus can be calculated using the atomic mass of thorium. The atomic mass of thorium is approximately 232.04 atomic mass units (u), which is equivalent to 3.856 x 10^-25 kg.
recoil speed of thorium nucleus ≈ (6.68 x 10^-27 kg)(2.10 x 10^7 m/s) / (3.856 x 10^-25 kg)
Evaluating this expression gives:
recoil speed of thorium nucleus ≈ 0.113 m/s
Therefore, the recoil speed of the thorium nucleus is approximately 0.113 m/s.