A neutron with a mass of 1.67x10^-27 kg traveling east with a kinetic energy of 2.00x10^-21J collides perfectly elastically with helium nucleus with a mass 6.68x10^-27 kg that is initially at rest. After the collision, the neutron has a kinetic energy 1.80x10^-21J. What angle was the neutron deflected during the collision?
To find the angle at which the neutron was deflected during the collision, we can use the concept of conservation of momentum and conservation of kinetic energy.
Step 1: Calculate the initial velocity of the neutron.
The kinetic energy of an object can be calculated using the formula: KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
Given that the kinetic energy of the neutron before the collision is 2.00x10^-21 J and its mass is 1.67x10^-27 kg, we can rearrange the formula to solve for velocity.
2.00x10^-21 J = (1/2)(1.67x10^-27 kg)v^2
v^2 = (2 * 2.00x10^-21 J) / (1.67x10^-27 kg)
v^2 = 2.40x10^6 m^2/s^2
v = √(2.40x10^6 m^2/s^2)
v ≈ 1549.59 m/s
Step 2: Calculate the final velocity of the neutron after the collision.
Given that the kinetic energy of the neutron after the collision is 1.80x10^-21 J, we can use the same formula as in Step 1 to calculate the final velocity.
1.80x10^-21 J = (1/2)(1.67x10^-27 kg)v^2
v^2 = (2 * 1.80x10^-21 J) / (1.67x10^-27 kg)
v^2 = 2.16x10^6 m^2/s^2
v = √(2.16x10^6 m^2/s^2)
v ≈ 1469.69 m/s
Step 3: Calculate the magnitude of the final velocity of the helium nucleus.
Since the helium nucleus is initially at rest, its final velocity magnitude will be equal to its initial velocity magnitude.
Given that the mass of the helium nucleus is 6.68x10^-27 kg, we can calculate the velocity using the formula: KE = (1/2)mv^2.
1.80x10^-21 J = (1/2)(6.68x10^-27 kg)v^2
v^2 = (2 * 1.80x10^-21 J) / (6.68x10^-27 kg)
v^2 = 5.39x10^5 m^2/s^2
v = √(5.39x10^5 m^2/s^2)
v ≈ 734.14 m/s
Step 4: Apply the law of conservation of momentum.
Before the collision, the total momentum of both particles is zero since the helium nucleus is initially at rest. Therefore, the total momentum after the collision must also be zero. Using this information, we can set up an equation:
(mass of neutron)(initial velocity of neutron) + (mass of helium nucleus)(initial velocity of helium nucleus) = (mass of neutron)(final velocity of neutron) + (mass of helium nucleus)(final velocity of helium nucleus)
(1.67x10^-27 kg)(1549.59 m/s) + (6.68x10^-27 kg)(0 m/s) = (1.67x10^-27 kg)(1469.69 m/s) + (6.68x10^-27 kg)(theta angle velocity)
theta angle velocity = [(1.67x10^-27 kg)(1549.59 m/s) - (1.67x10^-27 kg)(1469.69 m/s)] / (6.68x10^-27 kg)
theta angle velocity = (0.133x10^-22 Ns) / (6.68x10^-27 kg)
theta angle velocity ≈ 0.0199 m/s
Step 5: Calculate the angle of deflection.
We can use trigonometry to find the angle of deflection. The tangent of the angle of deflection can be calculated using the formula: tangent(angle) = (perpendicular velocity) / (initial velocity of neutron).
tangent(angle) = (theta angle velocity) / (1549.59 m/s)
angle = arctan([(theta angle velocity) / (1549.59 m/s)])
angle ≈ 0.76 degrees
Therefore, the neutron was deflected at an angle of approximately 0.76 degrees during the collision.
To find the angle by which the neutron was deflected during the collision, we can use the law of conservation of momentum and the law of conservation of kinetic energy.
Let's denote the initial velocity of the neutron as v1, the final velocity of the neutron as v1' (after collision), and the final velocity of the helium nucleus as v2 (after collision). We also need to find the angle θ by which the neutron is deflected.
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
(mass of neutron)(velocity of neutron) + (mass of helium nucleus)(velocity of helium nucleus) = (mass of neutron)(final velocity of neutron) + (mass of helium nucleus)(final velocity of helium nucleus)
(1.67x10^-27 kg)(v1) + (6.68x10^-27 kg)(0) = (1.67x10^-27 kg)(v1'cosθ) + (6.68x10^-27 kg)(v2cosθ)
Since the helium nucleus is initially at rest, its initial velocity is zero.
Next, we can use the law of conservation of kinetic energy, which states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision:
(1/2)(mass of neutron)(velocity of neutron)^2 = (1/2)(mass of neutron)(final velocity of neutron)^2 + (1/2)(mass of helium nucleus)(final velocity of helium nucleus)^2
(1/2)(1.67x10^-27 kg)(v1)^2 = (1/2)(1.67x10^-27 kg)(v1'^2) + (1/2)(6.68x10^-27 kg)(v2^2)
We know the initial kinetic energy is 2.00x10^-21 J, and the final kinetic energy of the neutron is 1.80x10^-21 J, so we can set up the equation:
(1/2)(1.67x10^-27 kg)(v1)^2 = (1/2)(1.67x10^-27 kg)(v1'^2) + (1/2)(6.68x10^-27 kg)(v2^2) - 2.00x10^-21 J
Now we have two equations with two unknowns (v1', v2), so we can solve them simultaneously to find these variables.
Once we have obtained the values of v1' and v2, we can use the following formula to find the angle θ:
tanθ = v2 / v1'
Finally, we can calculate the value of θ using the inverse tangent function.
Note: The calculations involved in solving the equations might be complicated. It is recommended to use a suitable numerical method or software to find the values accurately.