Question

You have been able to get a part time job with a medical physics group investigating ways to treat inoperable brain cancer. One form of cancer therapy being studied uses slow neutrons to knock a particle (either a neutron or a proton) out of the nucleus of the atoms which make up cancer cells. The neutron knocks out the particle it collides with in an inelastic collision. The rest of the nucleus does not move. After a single proton or neutron is knocked out of the nucleus, the nucleus decays, killing the cancer cell. To test this idea, your research group decides to measure the change in internal energy of a nitrogen nucleus after a neutron collides with one of the neutrons in its nucleus and knocks it out. The change in internal energy should be equal to the difference in kinetic energy before and after the collision. In the experiment, one neutron goes into the nucleus along the horizontal with a speed of 2.0 x 107 m/s. You detect two neutrons coming out, one at an angle of 30° above and the other 15° below the horizontal. You can now calculate the change in internal energy of the nucleus.

Im asking for a very detailed solution, including a physics diagram, the equations needed for the solution and the solution laid out. Please help, thank you.

Answer 1

Let the mass of a neutron be m

Before Collision:-

Let initial speed(given) of neutron be v = 2 * 10^7 m/s

Momentum = mv in right direction

and Momentum = 0 in upward direction

After Collision :-

Let speed of one neutron at angle of 30 deg be v1 m/s and other at angle of 15 deg be v2 m/s

Momentum = mv1 Cos 30 + mv2 Cos 15 in right direction

Momentum = mv1 Sin 30 - mv2 Sin 15 in upward direction

NOW BY Conservation of momentum principal in right direction, before and after collision momentum should be equal,

hence,

mv = mv1 Cos 30 + mv2 Cos 15 ...... [Ist equation]

Also in upward direction

0 =  mv1 Sin 30 - mv2 Sin 15 ...... [2nd equation]

ON SOLVING EUATION 1 AND EQUATION 2 WE GET

v1 = 2 v Sin 15 / (2 Sin15 Cos 30 + Cos 15) = 2 (v) 0.366 = 2 ( 2 * 10^7) 0.366 = 1.46 * 10^7 m/s

So v1 = 1.46 * 10^7 m/s

v2 = v / (2sin15Cos30 + Cos15) = v / 1.414 = 2* 10^7 /1.414 = 1.414 * 10^7 m/s

So v2 = 1.414 * 10^7 m/s

KE = Kinetic Energy

Now, Change in internal energy = KE ( before collision )- KE ( after collision )

KE(before) = 1/2 m v^2

m= mass of neutron = 1.6 * 10^-27 kg

v = 2 * 10^7 m/s so v^2 = 4 * 10^14 m/s

So, KE (Before) = 1/2 * (1.6*10^-27 )* (4*10^14) = 3.2 * 10^-13 Joules

KE(After collision) = 1/2 m(v1)^2 + 1/2 m (v2)^2 = 1/2 m [ (v1)^2 + (v2)^2 ]

= 1/2 (1.6 * 10^-27 kg) [ (1.46 * 10^7)^2 + (1.414 * 10^7)^2 ]

= 3.3 * 10^-13 Joules

Change in internal energy = KE ( before collision )- KE ( after collision )

= 3.2 * 10^-13 Joules - 3.3 * 10^-13 Joules = - 0.1 * 10^-13 Joules