In the first step of nuclear fission, a U-235 atom absorbs a neutron and becomes U-236 before becoming unstable and breaking into two smaller atoms and emitting various energies and particles. The mass of the U-235 is 235 amu and the mass of the neutron is 1.0 amu. If the uranium atom is moving to the right at 12,000 m/s and the neutron is moving to the left at 45,000 m/s before the collision, what will the velocity of the U-236 be immediately after the collision?
Well, let's see. We have an atom moving to the right and a neutron moving to the left. Looks like they're heading for a head-on collision! A classic case of atomic bumper cars, if you ask me.
So, when the atom and the neutron collide, they'll transfer some of their momentum to each other. Think of it like a game of pool, but with really tiny balls and no cool triangle thing at the beginning.
Now, to find the velocity of the U-236 atom after the collision, we'll need to do some momentum calculations. Grab your calculator, or maybe a slide rule if you're feeling retro.
To calculate momentum, we need to multiply the mass of an object by its velocity. Since the mass of the U-236 atom will be the same as the U-235 atom (235 amu), we can just worry about the velocities.
Given that the U-235 atom is moving to the right at 12,000 m/s and the neutron is moving to the left at 45,000 m/s, we can say that the total momentum before the collision is:
(U-235 momentum) + (neutron momentum) = 0
Why is the total momentum equal to zero, you ask? Well, because momentum is conserved in an isolated system, and we don't have any external forces acting on this atomic dance party.
So, to find the velocity of the U-236 atom after the collision, we'll set up an equation using the conservation of momentum:
(mass of U-236) * (velocity of U-236) + (mass of neutron) * (velocity of neutron) = 0
Now, plug in the values we know:
(235 amu) * (velocity of U-236) + (1.0 amu) * (-45,000 m/s) = 0
Solving for the velocity of U-236, we get:
(235 amu) * (velocity of U-236) = - (1.0 amu) * (-45,000 m/s)
(velocity of U-236) = - (1.0 amu) * (-45,000 m/s) / (235 amu)
And voila! You'll have the velocity of the U-236 atom after the collision. Just be sure to keep track of those units, so you don't end up with a velocity in furlongs per fortnight or something. Safety first!
To solve this problem, we need to apply the law of conservation of momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
Since momentum is a vector quantity, we need to consider the direction of motion for the uranium atom and the neutron. Let's assume that motion to the right is positive (+) and motion to the left is negative (-).
The momentum of an object is defined as the product of its mass and velocity:
Momentum = mass × velocity
Before the collision, the momentum of the U-235 atom is given by:
MomentumU-235 = massU-235 × velocityU-235
Before the collision, the momentum of the neutron is given by:
Momentumneutron = massneutron × velocityneutron
Assuming that momentum is conserved, the total momentum before the collision is:
Total momentum before = MomentumU-235 + Momentumneutron
The U-236 atom will immediately have a velocity after the collision, which we'll call velocityU-236.
The mass of U-236 is the same as the mass of U-235 since only a neutron was absorbed. Hence, the mass of U-236 is 235 amu.
According to the law of conservation of momentum, the total momentum after the collision is:
Total momentum after = massU-236 × velocityU-236
Setting the total momentum before and after the collision equal to each other, we get:
MomentumU-235 + Momentumneutron = massU-236 × velocityU-236
Now, let's plug in the given values:
massU-235 = 235 amu
velocityU-235 = 12,000 m/s (to the right, so positive)
massneutron = 1.0 amu
velocityneutron = -45,000 m/s (to the left, so negative)
massU-236 = 235 amu
MomentumU-235 = massU-235 × velocityU-235
= 235 amu × 12,000 m/s
MomentumU-235 = 2,820,000 amu m/s
Momentumneutron = massneutron × velocityneutron
= 1.0 amu × (-45,000 m/s)
Momentumneutron = -45,000 amu m/s
Now, let's substitute these values into the equation:
2,820,000 amu m/s + (-45,000 amu m/s) = 235 amu × velocityU-236
Simplifying the equation further:
2,775,000 amu m/s = 235 amu × velocityU-236
Now, let's solve for velocityU-236:
velocityU-236 = (2,775,000 amu m/s) / 235 amu
Calculating this:
velocityU-236 = 11,808.51 m/s
So, the velocity of the U-236 atom immediately after the collision is approximately 11,808.51 m/s.
To determine the velocity of the U-236 atom immediately after the collision, we need to apply the principle of conservation of momentum.
According to Newton's third law of motion, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object can be calculated by multiplying the mass of the object by its velocity. In this case, we have two objects: the U-235 atom and the neutron.
Given:
Mass of U-235 (m1) = 235 amu
Mass of neutron (m2) = 1.0 amu
Velocity of U-235 (v1) = 12,000 m/s (moving to the right)
Velocity of neutron (v2) = -45,000 m/s (moving to the left)
(Note: The negative sign indicates the opposite direction)
First, let's calculate the momentum before the collision:
Momentum before = (m1 * v1) + (m2 * v2)
Momentum before = (235 amu * 12,000 m/s) + (1.0 amu * -45,000 m/s)
Next, we'll find the total mass after the collision, which is the sum of the U-236 atom and the neutron:
Mass after = mass of U-236 (m1) + mass of neutron (m2)
Mass after = 236 amu + 1.0 amu
Now, we can calculate the velocity of the U-236 atom after the collision:
Momentum after = (mass after * velocity after)
Setting up the equation:
Momentum before = Momentum after
(m1 * v1) + (m2 * v2) = (mass after * velocity after)
Now, solve for velocity after:
Velocity after = [(m1 * v1) + (m2 * v2)] / mass after
Plug in the given values:
Velocity after = [(235 amu * 12,000 m/s) + (1.0 amu * -45,000 m/s)] / (236 amu + 1.0 amu)
Calculating:
Velocity after = (2,820,000 amu * m/s + -45,000 amu * m/s) / 237 amu
Velocity after = 2,775,000 amu * m/s / 237 amu
Velocity after = 11,700 m/s
Therefore, the velocity of the U-236 atom immediately after the collision is 11,700 m/s.
momentum is conserved
(235 * 12000) - (1 * 45000) = 236 * v