A neutron moving with velocity 3.96 x 107 m/s collides head on elastically with a helium atom moving with a velocity of 2.97 x 10 m/s in the opposite direction. (mass of helium = 4u). State and explain deductions one can make from this statement about momentum, energy and relative velocity. + (ii) Calculate the velocities of the neutron and helium after the collision.
From this statement, we can deduce several things:
1. Momentum: Since the collision is said to be elastic, we can conclude that the total momentum before the collision is equal to the total momentum after the collision. This means that the sum of the momenta of the neutron and the helium atom before the collision is equal to the sum of their momenta after the collision.
2. Energy: Elastic collisions are characterized by the conservation of kinetic energy. Therefore, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. This means the sum of the kinetic energies of the neutron and the helium atom before the collision is equal to the sum of their kinetic energies after the collision.
3. Relative velocity: The statement mentions that the neutron and the helium atom are moving in opposite directions. By colliding head-on, they will experience a change in relative velocity after the collision. The deduced information does not provide specific details about the change in relative velocity, but it can be inferred that the relative velocity will change due to the collision.
To calculate the velocities of the neutron and helium after the collision, we can use the principles of conservation of momentum and kinetic energy in an elastic collision.
Let v1 be the velocity of the neutron after the collision, and v2 be the velocity of the helium atom after the collision.
Using the law of conservation of momentum:
Momentum before collision = Momentum after collision
(mass of neutron) * (velocity of neutron before collision) + (mass of helium) * (velocity of helium before collision)
= (mass of neutron) * (velocity of neutron after collision) + (mass of helium) * (velocity of helium after collision)
(mass of neutron) * (3.96 x 10^7 m/s) + (mass of helium) * (-2.97 x 10 m/s)
= (mass of neutron) * (v1) + (mass of helium) * (v2)
Substituting the mass of helium as 4u and simplifying the equation:
(mass of neutron) * (3.96 x 10^7 m/s) - 4u * (2.97 x 10 m/s) = (mass of neutron) * (v1) + 4u * (v2)
Using the law of conservation of kinetic energy:
Kinetic Energy before collision = Kinetic Energy after collision
(0.5 * mass of neutron * (velocity of neutron before collision)^2) + (0.5 * mass of helium * (velocity of helium before collision)^2)
= (0.5 * mass of neutron * (velocity of neutron after collision)^2) + (0.5 * mass of helium * (velocity of helium after collision)^2)
(0.5 * mass of neutron * (3.96 x 10^7 m/s)^2) + (0.5 * 4u * (2.97 x 10 m/s)^2)
= (0.5 * mass of neutron * (v1)^2) + (0.5 * 4u * (v2)^2)
We have two equations with two unknowns (v1 and v2), which can be solved simultaneously to find their values.
From the given statement, we can make the following deductions about momentum, energy, and relative velocity:
1. Momentum: In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. Therefore, the total momentum of the neutron and helium before the collision is zero since they have equal but opposite velocities.
2. Energy: In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Therefore, the total kinetic energy of the neutron and helium before the collision is also zero since they have equal but opposite velocities.
3. Relative Velocity: The relative velocity of the two particles before the collision is the sum of their individual velocities. In this case, the neutron and helium are moving towards each other, so the relative velocity before the collision is the sum of their magnitudes.
Now, to calculate the velocities of the neutron and helium after the collision, we can use the principles of conservation of momentum and conservation of energy:
Let the velocities of the neutron and helium after the collision be V1 and V2, respectively.
According to the conservation of momentum:
Momentum before the collision = Momentum after the collision
(mass of the neutron) x (velocity of the neutron) + (mass of the helium) x (velocity of the helium) = (mass of the neutron) x (V1) + (mass of the helium) x (V2)
Since the mass of the neutron is much greater than the mass of the helium, its velocity will be significantly reduced while the helium atom will experience a small change in its velocity.
Now, applying the conservation of kinetic energy:
Kinetic energy before the collision = Kinetic energy after the collision
(0.5 x mass of the neutron x (velocity of the neutron)^2) + (0.5 x mass of the helium x (velocity of the helium)^2) = (0.5 x mass of the neutron x (V1)^2) + (0.5 x mass of the helium x (V2)^2)
Substituting the given values, we can solve these equations to calculate the velocities V1 and V2 of the neutron and helium after the collision.