A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times that of the neutron. The helium nucleus is observed to rebound at an angle è'2 = 45° from the neutron's initial direction. The neutron's initial speed is 6.6x10^5 m/s. Determine the angle at which the neutron rebounds, è'1, measured from its initial direction.
___°
What is the speed of the neutron after the collision?
___ m/s
What is the speed of the helium nucleus after the collision?
___ m/s
Elastic? Ok, momentum equations and energy equations can be used.
The sum of the final KE =initial KE
The sum of momentums in orig direction=momentum of neutron originally.
The sum of momentum in direction perpendicular to orig direction=zero
You will have three unknowns:
velocity of neutron after
velocity of He after
Theta.
Three equations, three unknowns. A little algebra and trig :)
thanks but I know that there are three unknowns and three equations. Getting the formulas are the difficult part.
To solve this problem, we can use the laws of conservation of momentum and conservation of kinetic energy.
1. First, let's denote the initial direction of the neutron as the x-axis and the direction perpendicular to it as the y-axis. Since the helium nucleus is initially at rest, its momentum before the collision is zero.
2. Conservation of momentum: Before the collision, the total momentum in the x-direction is given by the momentum of the neutron: P1x = m1 * v1, where m1 is the mass of the neutron and v1 is its initial velocity. The total momentum in the y-direction is also zero.
After the collision, the total momentum in the x-direction is given by the momentum of the neutron and helium nucleus in their respective rebound directions: P2x = m1 * v2x + m2 * v2x', where v2x and v2x' are the x-components of the velocities of the neutron and helium nucleus after the collision respectively. The total momentum in the y-direction is given by the y-components of the velocities: P2y = -m1 * v2y.
Since the total momentum in the y-direction is zero, we have: P2y = - m1 * v2y = 0, which implies that v2y = 0. The helium nucleus rebounds perpendicular to the initial direction, so it doesn't have a y-component of velocity.
Thus, the conservation of momentum equations become:
m1 * v1 = m1 * v2x + m2 * v2x' (1)
0 = - m1 * v2y = 0
3. Conservation of kinetic energy: The total kinetic energy before the collision is given by:
KE1 = (1/2) * m1 * v1^2
After the collision, the total kinetic energy is given by:
KE2 = (1/2) * m1 * v2^2 + (1/2) * m2 * v2'^2, where v2 and v2' are the magnitudes of the velocities of the neutron and helium nucleus after the collision, respectively.
As this is an elastic collision, the total kinetic energy is conserved, so we have: KE1 = KE2, which implies:
(1/2) * m1 * v1^2 = (1/2) * m1 * v2^2 + (1/2) * m2 * v2'^2 (2)
4. Now let's solve equations (1) and (2) to find the values of v2 and v2'.
First, from equation (1):
m1 * v1 = m1 * v2x + m2 * v2x' (3)
Similarly, from equation (2):
v1^2 = v2^2 + v2'^2 (4)
5. We have two unknowns, v2 and v2', and two equations (3) and (4). We can solve for v2 and v2' by substituting the given values:
m1 = mass of the neutron = 1 neutron mass = 1.675 x 10^-27 kg
m2 = mass of the helium nucleus = 4 times the neutron mass = 4 * 1.675 x 10^-27 kg
v1 = initial velocity of the neutron = 6.6 x 10^5 m/s
Substituting these values into equations (3) and (4), we can solve for v2 and v2':
Equation (3):
(1.675 x 10^-27 kg) * (6.6 x 10^5 m/s) = (1.675 x 10^-27 kg) * v2x + (4 * 1.675 x 10^-27 kg) * v2x'
10^-27 kg * m/s = v2x + 4 * v2x'
10^-27 kg * m/s = v2x * (1 + 4)
Therefore, v2x = (10^-27 kg * m/s) / 5
Equation (4):
(6.6 x 10^5 m/s)^2 = v2^2 + v2'^2
v2^2 = (6.6 x 10^5 m/s)^2 - v2'^2
6. Now we can solve for v2 and v2' using the above values.
To find the angle at which the neutron rebounds (è'1):
tan(è'1) = v2y / v2x = 0 / v2x = 0
Therefore, è'1 = 0 degrees.
The speed of the neutron after the collision:
v2 = sqrt[(6.6 x 10^5 m/s)^2 - v2'^2]
The speed of the helium nucleus after the collision:
v2' = sqrt[(10^-27 kg * m/s)^2 / 5^2] = sqrt[(10^-54 kg^2 * m^2/s^2) / 25]
Please substitute the given values to calculate the final answers for the speed of the neutron after the collision and the speed of the helium nucleus after the collision.