# Finding Speed of Neutron after Decay

## Homework Statement

A Helium 5 at rest decays into Helium 4 and a neutron.
Mass of Helium 4 = 6.648 x 10^-27
Mass of Neutron = 1.67493 x 10^-27
Helium 4 has a momentum of 1.903 x10^-20 and Ek 2.723x10^-14

## Homework Equations

E=mc^2
v= square root of (E2/m)

## The Attempt at a Solution

I have found the mass defect for both Helium 4 and the neutron. and Helium 5.
I think i can use E=mc^2, and then use E in v= square root of (E2/m). I am stuck in trying to figure out what I should use for m? The mass defect on the before or after the decay? Any help appreciated.

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haruspex
Homework Helper
Gold Member
Hi Bob,

This forum is fairly generic across introductory physics, so, like me, many of those who might help you are not familiar with atomic physics terminology, conventions, and equations. E.g., what is E2? If you could avoid or explain such usages, and post all equations that may be relevant, you are more likely to get help.

Hi Bob,

This forum is fairly generic across introductory physics, so, like me, many of those who might help you are not familiar with atomic physics terminology, conventions, and equations. E.g., what is E2? If you could avoid or explain such usages, and post all equations that may be relevant, you are more likely to get help.

Hello. I will add some clarification

I manipulated the formula for kinetic energy, which is E=(1/2)mv^2
E is energy, m is mass, and v and is velocity.

haruspex
Homework Helper
Gold Member
Hello. I will add some clarification

I manipulated the formula for kinetic energy, which is E=(1/2)mv^2
E is energy, m is mass, and v and is velocity.
Ah, E2 meant 2Ek, right?
But there are two masses resulting, and they will move at different speeds. For the neutron (at least) that speed may be relativistic, so you might need a post-Newtonian equation for its energy. You'll need to use momentum conservation to pin down the two speeds.

Ah, E2 meant 2Ek, right?
But there are two masses resulting, and they will move at different speeds. For the neutron (at least) that speed may be relativistic, so you might need a post-Newtonian equation for its energy. You'll need to use momentum conservation to pin down the two speeds.
Yes it means 2Ek. The neutron and He4 move off so their speed would be different. OK, i was unsure if the conservation of momentum law would apply here. I tried to use E=mc^2 to find E, and then inserting E into my formula for v but I think the answer is wrong for the velocity of the neutron...

haruspex
Homework Helper
Gold Member
Yes it means 2Ek. The neutron and He4 move off so their speed would be different. OK, i was unsure if the conservation of momentum law would apply here. I tried to use E=mc^2 to find E, and then inserting E into my formula for v but I think the answer is wrong for the velocity of the neutron...
You will need to post all your working if you want anyone to spot where you are going wrong. Preferably, use LaTeX.

have you tried a simple momentum balance?

If the momentum of helium 5 is zero and the momentum afterwards of the helium 4 is given (due to the recoil of ejecting the neutron), you can work out the momentum of the neutron - the sum momentum is conseved.

If you are getting speeds of 10% or more of the speed of light, you might want to use the relativistic momentum equation. But I think the % difference will be not worth worrying about.

have you tried a simple momentum balance?
.
Like this?

$$p_{H4} = p_{n}$$

$$v_{x} = p_{x} / m{x)$$

$$v_{H4} = 2,862,515 v_{n} = 11,361,669$$

No units were given... m is in (kg)? momentum is in (kg-m/s)? for m/s ?

haruspex
Homework Helper
Gold Member
Like this?

$$p_{H4} = p_{n}$$

$$v_{x} = p_{x} / m{x)$$

$$v_{H4} = 2,862,515 v_{n} = 11,361,669$$

No units were given... m is in (kg)? momentum is in (kg-m/s)? for m/s ?
Where did you get the 2,862,515 from? If you insist on working with numerics it makes it very hard for others to follow what you are doing. Please learn to do everything symbolically until the final step.
As I posted before, you know the total energy released but it will be shared between the two bodies. You need to use conservation of energy and conservation of momentum to find the two speeds. Create an unknown for each speed and write down the conservation equations. And as I mentioned, you will probably need to use the relativistic energy expression for the lighter body. ##\frac 12 m_nv_n^2## might not be good enough.

Like this?No units were given... m is in (kg)? momentum is in (kg-m/s)? for m/s ?
The units are SI but don't worry about that yet.

You know (or will make the assumption) that the total momentum is zero.

You know the momentum of the helium 4 as it recoils, so you know

phelium4 + pneutron = 0

The units are SI but don't worry about that yet.

You know (or will make the assumption) that the total momentum is zero.

You know the momentum of the helium 4 as it recoils, so you know

phelium4 + pneutron = 0
Right, that was the first equation.

Second equation was the classic momentum equation solved for v

He said he had one of the momentums, so it took the provided momentum and divided it by the two masses.

DEvens
Gold Member

## Homework Statement

A Helium 5 at rest decays into Helium 4 and a neutron.
Mass of Helium 4 = 6.648 x 10^-27
Mass of Neutron = 1.67493 x 10^-27
Helium 4 has a momentum of 1.903 x10^-20 and Ek 2.723x10^-14

## Homework Equations

E=mc^2
v= square root of (E2/m)

## The Attempt at a Solution

I have found the mass defect for both Helium 4 and the neutron. and Helium 5.
I think i can use E=mc^2, and then use E in v= square root of (E2/m). I am stuck in trying to figure out what I should use for m? The mass defect on the before or after the decay? Any help appreciated.
Some suggestions:

First, always include your units. You have dropped the units for the He4 momentum. You have dropped the units for the He4 energy. This is vital, since none of the rest of the question makes sense without them.

Make your equations look simple. You wrote E2 and later folks asked you, what is E2. You should have written 2 Ek, or better, ## 2E_k##.

I look and look, but I do not see anywhere in your problem statement what it is you are meant to determine?

Do a ball-park sitting-down kind of estimate of things to see if you are getting things correct. The He4 is roughly four times the mass of the neutron. So it has to be going roughly one quarter as fast. And since the neutron is going 4 times as fast but has one quarter the mass, it has to have four times the kinetic energy. Roughly.

I look and look, but I do not see anywhere in your problem statement what it is you are meant to determine?

Its in the title "Finding Speed of Neutron after Decay"

DEvens
Gold Member
I look and look, but I do not see anywhere in your problem statement what it is you are meant to determine?

Its in the title "Finding Speed of Neutron after Decay"
Indeed. But it is still not in the problem statement where it belongs.

He said he had one of the momentums, so it took the provided momentum and divided it by the two masses.

Helium 4 is 2 protons, 2 neutrons, so mass is = 4 amu

neutron mass = 1 amu

Momentum is conserved, so I am expecting that the neutron is going 4x faster than the helium 4, but in the opposite direction. And you know the speed of the helium 4, because you know its momentum

Indeed. But it is still not in the problem statement where it belongs.
ah! it is not...

Wow everyone. Really appreciate the help with this one.

Here is a link with the full question and my attempt. There is a lot of info in the question and I am getting thrown off by that tbh.

http://imgur.com/aHxilFE

Wow everyone. Really appreciate the help with this one.

Here is a link with the full question and my attempt. There is a lot of info in the question and I am getting thrown off by that tbh.

http://imgur.com/aHxilFE
Could someone please help me understand by what is meant with " the energy equivalence of the mass defect is observed as an increase in the systems kinetic energy"?

have you tried a simple momentum balance?

If the momentum of helium 5 is zero and the momentum afterwards of the helium 4 is given (due to the recoil of ejecting the neutron), you can work out the momentum of the neutron - the sum momentum is conseved.

If you are getting speeds of 10% or more of the speed of light, you might want to use the relativistic momentum equation. But I think the % difference will be not worth worrying about.
Yup this is what I tried!

## Homework Statement

A Helium 5 at rest decays into Helium 4 and a neutron.
Mass of Helium 4 = 6.648 x 10^-27
Mass of Neutron = 1.67493 x 10^-27
Helium 4 has a momentum of 1.903 x10^-20 and Ek 2.723x10^-14

## Homework Equations

E=mc^2
v= square root of (E2/m)

## The Attempt at a Solution

I have found the mass defect for both Helium 4 and the neutron. and Helium 5.
I think i can use E=mc^2, and then use E in v= square root of (E2/m). I am stuck in trying to figure out what I should use for m? The mass defect on the before or after the decay? Any help appreciated.

The next part of question asks me to find the mass of the Helium nucleus.

Last edited:
Ah, E2 meant 2Ek, right?
But there are two masses resulting, and they will move at different speeds. For the neutron (at least) that speed may be relativistic, so you might need a post-Newtonian equation for its energy. You'll need to use momentum conservation to pin down the two speeds.
http://imgur.com/aHxilFE

I posted my solution here if you have a chance to take a look.
Cheers.

haruspex
Homework Helper
Gold Member
Could someone please help me understand by what is meant with " the energy equivalence of the mass defect is observed as an increase in the systems kinetic energy"?
The resulting He4 and neutron will have a total rest mass less than that of the He5. The lost mass is the 'mass defect'. This will be equal to the total KE of the two particles. I thought you understood that, from what you wrote in the initial post.
http://imgur.com/aHxilFE

I posted my solution here if you have a chance to take a look.
Cheers.
That's too messy and incomplete to follow/review. Please don't post algebra as images. Take the trouble to type in your working, preferably in LaTeX.
Start with stating the relativistic formulae for momentum and energy in terms of rest mass and velocity. Create variable names to represent the the rest masses and velocities of the neutron and the He4, and the mass defect. Also create variable names for the given values, the momentum and energy of the He4. Then apply the standard formulae to get equations involving your variables.
Do not plug in numeric values at all, yet.

Last edited:
BvU
The next part of question asks me to find the mass of the Helium nucleus.
that is in the data

What is the speed is of the helium 4 nucleus? You know its momentum, you know its mass.

What is the speed is of the neutron? You know its momentum, you know its mass.

that is in the data

What is the speed is of the helium 4 nucleus? You know its momentum, you know its mass.

What is the speed is of the neutron? You know its momentum, you know its mass.
Sorry I meant to say the questions ask what the mass of a Helium 5 nucleus is.

The resulting He4 and neutron will have a total rest mass less than that of the He5. The lost mass is the 'mass defect'. This will be equal to the total KE of the two particles. I thought you understood that, from what you wrote in the initial post.

That's too messy and incomplete to follow/review. Please don't post algebra as images. Take the trouble to type in your working, preferably in LaTeX.
Start with stating the relativistic formulae for momentum and energy in terms of rest mass and velocity. Create variable names to represent the the rest masses and velocities of the neutron and the He4, and the mass defect. Also create variable names for the given values, the momentum and energy of the He4. Then apply the standard formulae to get equations involving your variables.
Do not plug in numeric values at all, yet.
Formula for momentum is p=mv (m=mass, v= velocity)
p=momentum
Conservation of momentum of law states that total p before reaction = total p after reaction
p before = 0 N (newtons)
p after = (Helium 4 = 1.90306 x 10-20N) + ( momentum of neutron)
Since "total p before must equal total p after ", the momentum of the neutron must be: (-)1.90306 x 10-20N.
If we add the momentum of He4 and of the Neutron after the reaction we get:
(1.90306 x 10-20N) + (- 1.90306 x 10-20N)
= 0
Therefore the laws of conservation of momentum are obeyed.
I manipulated the formula p=mv to solve for velocity, the resulting formula is v=(p/m)
So the velocity of the neutron should be
v=(p/m)
v=(- 1.90306 x 10-20N )/ (1.67493 x 10-27 kg)
v= 1.14x107 m/s

I cant get latex to work on my pc for some reason. Any help would be appreciated at this point. The next part of the question asks to find the mass of the helium 5 nucleus and I am puzzled on that one.
Thanks.