A uranium nucleus (mass 238 u), initially at rest, undergoes radioactive decay. After an alpha particle (mass 4.0 u) is emitted, the remaining nucleus is thorium (mass 234 u). If the alpha particle is moving at 0.055 the speed of light, what is the recoil speed of the thorium nucleus?
To find the recoil speed of the thorium nucleus, we can use the law of conservation of momentum. According to this law, the total momentum before the decay must be equal to the total momentum after the decay.
1. Start by calculating the initial momentum of the uranium nucleus. Since it is initially at rest, its momentum is zero (p_initial = 0).
2. Next, calculate the momentum of the alpha particle after it is emitted. We know the mass of the alpha particle is 4.0 u, and it is moving at 0.055 times the speed of light (c). The relativistic momentum equation is p_alpha = γm_alpha v_alpha, where γ is the Lorentz factor, m_alpha is the mass of the alpha particle, and v_alpha is its velocity.
To find γ, we use the equation γ = 1/sqrt(1 - (v_alpha/c)^2).
Plugging in the values, we get:
γ = 1/sqrt(1 - (0.055c/c)^2) = 1/sqrt(1 - 0.055^2) = 1/sqrt(1 - 0.003025) ≈ 1.000052915.
Now we can calculate the momentum of the alpha particle:
p_alpha = γm_alpha v_alpha = 1.000052915 * 4.0u * 0.055c ≈ 0.2200184u*c.
3. Since momentum is conserved, the total momentum after the decay must equal the initial momentum (since the uranium nucleus was initially at rest).
p_after = p_alpha + p_thorium = 0.
Therefore, we can solve for the momentum of the thorium nucleus:
p_thorium = -p_alpha = -0.2200184u*c.
4. Finally, we can calculate the recoil speed of the thorium nucleus. We'll use the equation:
p_thorium = m_thorium * v_thorium, where m_thorium is the mass of the thorium nucleus, and v_thorium is its recoil speed.
Rearranging the equation to solve for v_thorium, we get:
v_thorium = p_thorium / m_thorium.
Plugging in the values, we obtain:
v_thorium = (-0.2200184u*c) / 234u ≈ -0.0009403c.
Therefore, the recoil speed of the thorium nucleus is approximately -0.0009403 times the speed of light (c). Note that the negative sign indicates that the recoil speed is in the opposite direction of the alpha particle's motion.
To calculate the recoil speed of the thorium nucleus, we can use the principle of conservation of momentum. In this case, we assume that the system is isolated, so the initial momentum is zero.
The initial momentum of the system (before decay) is given by:
P_initial = 0
After the alpha particle is emitted, the remaining uranium nucleus recoils in the opposite direction. The momentum of the alpha particle (final momentum) is given by:
P_alpha = m_alpha * v_alpha
Where:
m_alpha = mass of the alpha particle = 4.0 u
v_alpha = velocity of the alpha particle = 0.055c (c: speed of light)
The momentum of the uranium nucleus (remaining nucleus) can be calculated by the principle of conservation of momentum:
P_uranium + P_alpha = 0
Since the uranium nucleus is initially at rest, its momentum is:
P_uranium = 0
Therefore, we can write:
P_alpha = -P_uranium
Using the given masses, we can express the momentum in terms of velocities using the relation:
momentum = mass * velocity
P_alpha = (m_alpha * v_alpha) = (-m_uranium * v_uranium)
We can find the recoil speed of the thorium nucleus (v_thorium) by rearranging and solving the equation:
(m_uranium * v_uranium) = (-m_thorium * v_thorium)
Rearranging the equation, we get:
v_thorium = -(m_uranium * v_uranium) / m_thorium
where:
m_uranium = mass of the uranium nucleus = 238 u
v_uranium = velocity of the uranium nucleus (initially at rest) = 0
m_thorium = mass of the thorium nucleus = 234 u
Plugging in the known values, we get:
v_thorium = - (238 * 0) / 234
Since the first term on the numerator is zero, the recoil speed of the thorium nucleus (v_thorium) is zero.
Hence, the recoil speed of the thorium nucleus after the alpha particle is emitted is zero.