A 238U nucleus is moving in the x-direction at 5.0x10^5m/s when it decays into an alpha particle (4He) and a 234Th nucleus. The alpha moves at 1.4x10^7 m/s at 22 degrees above the x-axis. Find the recoil velocity of the thorium.

I know that the original x momentum is 238*5.0*10^5 which is equal to 1.19x10^8 and the original y momentum is 0. I'm confused where to go after this.

momentum is conserved

find the momenta in x and y for the two decay products

the two y momenta add to zero, and the two x momenta equal the original

add the vectors to find the 234Th velocity

To find the recoil velocity 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.

Before the decay:
The uranium nucleus has momentum in the x-direction only, given by 238 * 5.0 * 10^5 = 1.19 * 10^8 kg*m/s.
The alpha particle has momentum in the x and y-directions, given by:
px_alpha = m_alpha * v_alpha * cos(theta) --> (1)
py_alpha = m_alpha * v_alpha * sin(theta) --> (2)

After the decay:
The alpha particle has momentum in the x-direction only, given by m_alpha * v_alpha. This momentum is equal in magnitude but opposite in direction to the uranium nucleus's momentum.
The thorium nucleus has momentum in the x and y-directions, given by:
px_recoil = m_thorium * v_recoil * cos(phi) --> (3)
py_recoil = m_thorium * v_recoil * sin(phi) --> (4)

According to conservation of momentum, the total momentum before the decay is equal to the total momentum after the decay:
1.19 * 10^8 + 0 = - (m_alpha * v_alpha) + (m_thorium * v_recoil * cos(phi))

Now, let's substitute the known values into these equations:
m_alpha = 4 amu = 4 * 1.66 * 10^-27 kg (mass of an alpha particle)
v_alpha = 1.4 * 10^7 m/s (velocity of the alpha particle)
phi = 22 degrees (angle above the x-axis)

Plugging in these values, we get:
1.19 * 10^8 = - (4 * 1.66 * 10^-27) * (1.4 * 10^7) + (m_thorium * v_recoil * cos(22))

Now, let's solve this equation for v_recoil.

First, multiply the mass and velocity terms for the alpha particle:
- (4 * 1.66 * 10^-27) * (1.4 * 10^7) = - (5.92 * 10^-20) kg*m/s

Next, subtract this value from both sides of the equation:
1.19 * 10^8 + (5.92 * 10^-20) kg*m/s = m_thorium * v_recoil * cos(22)

Now, divide both sides by (m_thorium * cos(22)):
v_recoil = (1.19 * 10^8 + (5.92 * 10^-20) kg*m/s) / (m_thorium * cos(22))

Since we have not been provided with the mass of the thorium nucleus (m_thorium), we cannot calculate the recoil velocity of the thorium nucleus without that information.