Neutron, proton collision problem

In summary, the problem involves using the conservation of momentum and kinetic energy to solve for the maximum velocity and relative mass of a neutron in an experiment involving collisions with hydrogen and nitrogen atoms. The equations for the maximum velocity and relative mass are derived from the given information and the concept of elastic collisions. The equation in part (c) is obtained by eliminating z between two equations.
  • #1
ofeyrpf
30
0
The following problem appeared on the A2 Edexcel Physics unit 4 exam paper January 2012 question 18. The solution, as given by the exam board, is attached.

Question:
18. James Chadwick is credited with discovering the neutron in 1932.

Beryllium was bombarded with alpha particles, knocking neutrons out of the beryllium atoms. Chadwick placed various targets between the beryllium and a detector. Hydrogen and nitrogen atoms were knocked out of the targets by the neutrons and the kinetic energies of these atoms were measured by the detector.

(a) The maximum energy of a nitrogen atom wa found to be 1.2 MeV.

Show that the maximum velocity of the atom is about 4 x 106 m/s.

mass of nitrogen atom = 14u, where u = 1.66 x 10-27 kg


Solution:
The set up as I understand it is,
alpha --> Be --> neutron --> target --> Ni or H --> detector

v = sqrt(2(1.2x10-6x1.6x10-19)/(14x1.66x10-19)) = 4.06x10-6 m/s No problems here.

Question
(b)The mass of a neutron is Nu (where N is the relative mass of the neutron) and its initial velocity is x. the nitrogen atom, mass 14u, is initially stationary and is then knocked out of the target with a velocity, y, by a collision with a neutron.

(i) Show that the velocity, z, of the neutron after the collision can be written as

z = (Nx - 14y)/N​

Solution:
momentum before = momentum after

Nux = 14uy - Nuz

rearranging gives,

z = (Nx - 14y)/N No problems here.

Question
(ii)The collision between this neutron and the nitrogen atom is elastic. What is meant by an elastic collision?

Solution
In an elastic collision the kinetic energy is conserved. No problems here.

Question
(iii) Explain why the kinetic energy Ek of the nitrogen atom is given by

Ek = (Nu(x2 - z2)/(2)​


Solution:
Using conservation of kinetic energy,

EK(n) = Ek(Ni) + Ek(n)

(1/2)Nux2 = (1/2)14Nuy2 = (1/2)Nuz2

y2 = (x2 - y2)/(14)

Ek(Ni) = (1/2)14Nuy2

= (1/2)(14Nu(x2 - z2)/14)

= (Nu(x2 - z2)/(2)

For this calculation to work, the mass of the Ni has to be 14Nu but in the question it is given as 14u. That is the first thing I don't understand.

Question
(c) The two equations in (b) can be combined and z can be eliminated to give

y = (2Nx)/(N + 14)​

Solution
The question does not ask how this is done but I'd like to know and can't figure it out. I tried substituting

z = (Nx - 14y)/(N) into Ek = (Nu(x2 - z2))/(2)

and this gives,

(2Ek)/(Nu) = x2 - ((Nx - 14y)/(N))2

But this has an Ek in it, so I don't see how to get to the required y = (2nx)/(N + 14) This is the second problem I have, not understanding where this equation comes from.

Question
(i) The maximum velocity of hydrogen atoms knocked out by neutrons in the same experiment was 30 x 107 m/s. The mass of a hydrogen atom is 1u.

Show that the relative mass N of the neutron is 1.


Solution
There is an error in the question here. Instead of 30 x 107 m/s it should 3.0 x 107 m/s.

The equation given in the question applies to Nitrogen and can be rearranged to give

2Nx = yNi(N + 14) = 4.1 x 106Ni(N + 14)

yNi = 4.1 x 106 m/s. This is obtained from part (a).

For hydrogen then

2nx = yH(N + 1) = 3.0 x 107(N+1)

These two equation can be combined giving,

4.1 x 106Ni(N + 14) = 3.0 x 107(N+1)

from which N can be solved

N = (3 x 107-14 x 4.1 x 106)/(4.1 x 106 - 3 x 107)
= 1.05 which is approximately 1

Question
(ii) This equation can not be applied to all collisions in this experiment. Suggest why.

Solution
As the atoms approach the speed of light their mass does not remain constant, it increases.
 

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  • #2
ofeyrpf said:
(1/2)Nux2 = (1/2)14Nuy2 = (1/2)Nuz2
Shouldn't that read:
(1/2)Nux2 = (1/2)14uy2 + (1/2)Nuz2
?
 
  • #3
Neutron, Ni, H, collision problem

Hi haruspex,

Thanks for your reply. Looking back on this post I can't believe how many typos I've made.

Yes you are correct it should read...

18. (b) (iii)

EK(n) = Ek(Ni) + Ek(n)

(1/2)Nux2 = (1/2)14uy2 + (1/2)Nuz2

14uy2 = Nu(x2 - z2)

y2 = N(x2 - z2)/14

Ek(Ni) = (1/2)14uy2

= (1/2)(14uN(x2 - z2)/14)

= Nu(x2 - z2)/(2)

So that is correct then, thanks.

Now if I could just figure out where they get the equation in part (c), I'd be happy.
 
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  • #4
For c, you have two equations:
Nx2=14y2+Nz2; Nz = Nx - 14y
Just eliminate z between them.
 
  • #5
$$
Nx^2=14y^2+Nz^2\\
\mbox{and}\\
Nz=Nx-14y\\
\mbox{so,}\\
z^2=\frac{(Nx-14y)(Nx-14y)}{N^2}\\
=\frac{N^2x^2-28yNx+196y^2}{N^2}\\
\mbox{substituting this into the first equation gives,}\\
Nx^2=14y^2+N\left(\frac{N^2x^2-28yNx+196y^2}{N^2}\right)\\
0=Ny^2-2yNx+14y^2\\
2Nx=y(N+14)\\
y=\frac{2Nx}{n+14}
$$
Thanks for your help,

Shane
 
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  • #6
blah
 

FAQ: Neutron, proton collision problem

What is a neutron-proton collision problem?

The neutron-proton collision problem refers to the challenge of understanding and predicting the behavior of particles during collisions between neutrons and protons, which are subatomic particles found in the nucleus of an atom. This problem is important in fields such as nuclear physics and particle physics, as it helps us understand the fundamental interactions between these particles.

Why is the neutron-proton collision problem important?

The neutron-proton collision problem is important because it helps us understand the behavior and properties of the atomic nucleus, which is the central part of an atom. It also provides insight into the strong nuclear force, which is responsible for binding the protons and neutrons together in the nucleus. This understanding is crucial for advancements in nuclear energy and particle accelerator technology.

How do scientists study neutron-proton collisions?

Scientists study neutron-proton collisions by using powerful particle accelerators, such as the Large Hadron Collider, to accelerate particles to very high speeds and then collide them with each other. They also use advanced detectors to measure the particles produced during the collisions and analyze the data to gain insights into the behavior of the particles.

What challenges are associated with the neutron-proton collision problem?

The neutron-proton collision problem presents several challenges for scientists. One of the main challenges is the complexity of the interactions between the particles, which can be difficult to accurately model and predict. Additionally, the high energies and short time scales involved in these collisions make them challenging to study and require advanced technology and techniques.

How does understanding neutron-proton collisions contribute to our knowledge of the universe?

Understanding neutron-proton collisions is crucial for our knowledge of the universe as it helps us understand the building blocks of matter and the fundamental forces that govern the behavior of particles. This knowledge also contributes to our understanding of the origins and evolution of the universe, as well as the structure and behavior of matter at a microscopic level.

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