Trying to solve conservation of momentum problem

[patric44]

Homework Statement

its a nuclear physics problem :
a deuterium atom with energy 4Mev collides with a boron atom B5/10 in an elastic collision
producing an H1/1 and B5/11 ,the Q of the reaction = 9.23 Mev .
find the E,E' , the energy of the boron and the hydrogen
the unknowns :
E,E', theta of boron.

Homework Equations

the solution is in solving these three equation :

The Attempt at a Solution

I first tried to get the third equation by eliminating the angle $\theta$
then when trying to solve those three equations I get:

 

[patric44]

guys i made a little mistake the third equation in my attempt sulould be 22E=16+2E'+... not (2E')^0.5
and for the other three papers also. you will understand when trying to solve it .

 

[patric44]

need help guys

 

[haruspex]

Your first page is ok, but you lose your way in trying to eliminate θ. What simple algebraic relationship is there between sin and cos?

 

[patric44]

here another attempt i almost done but it stil complicated ( i had solved millions of equations ODE , PDE , vector calculus and many) but this
still got me :

 

[haruspex]

here another attempt i almost done but it stil complicated ( i had solved millions of equations ODE , PDE , vector calculus and many) but this
still got me :
View attachment 197341 View attachment 197342

Please try to answer my question in post #4.

 

[patric44]

there is a lot of trigonometric identities but i don't know which one to use :
the one that i used in getting the third equation (sin^2+cos^2=1)
those three equation is correct i checked it :

may be i sould use :
(cos2theta = 1-2sin^theta)

Please try to answer my question in post

do you have particular one in mind

 

[gneill]

I suspect that your set of three equations are not independent, and I'm rather unclear on how you arrived at the third equation. Some annotation of your math would be nice. I also don't see where you've incorporated the Q of the nuclear reaction. Surely that must be an important contribution to the post-collision KE.

 

[patric44]

I suspect that your set of three equations are not independent, and I'm rather unclear on how you arrived at the third equation. Some annotation of your math would be nice. I also don't see where you've incorporated the Q of the nuclear reaction. Surely that must be an important contribution to the post-collision KE.

first thanks for responding
what you mean that the equations is not independant ?

in the book that contain that problem (which is a very sick book) has a very
breif solution goes like that :
1-by squaring and summing and eliminating theta we get ( the third equation)
2- by substituting in 1,2 equations we get (E=3.06 and E' = 10.17)
just like that (not a single step)

the third equation derivation i came up with my self :

the Q also mentiend in the book that its equal 9.23Mev
my only problem is in solving those three equations

 

[gneill]

the third equation derivation i came up with my self :

So it is based on the two existing momentum equations, hence it contains no new information. That means it is entirely dependent on the other two equations. The set of three equations is not a set of independent equations, and you can't use them to solve for all three variables.

You need to incorporate the Q information. You've written two momentum equations and that's all the information you can get from that. What other conservation law can you apply?

 

[patric44]

but in the book its said by subsituting of the third equation in 1,2
i forgot to mention that :
E+E'= 4+9.23 = 13.23Mev
which i tried to sole in it but got wrong answers

 

[gneill]

but in the book its said by subsituting of the third equation in 1,2

I suspect that the book is incorrect on that point, and that was not how the problem was solved.

i forgot to mention that :
E+E'= 4+9.23 = 13.23Mev
which i tried to sole in it but got wrong answers

I think what the book's author actually did was use the third equation (which eliminated θ) and the energy equation above to solve for E and E'. Then θ can be found using either of the first two equations.

 

[patric44]

i think some thing is wrong in the third equation in the book
becouse i tried to plug E' = 10.17 i don't get E = 3.06 as mentiend in the book

 

[gneill]

i think some thing is wrong in the third equation in the book
becouse i tried to plug E' = 10.17 i don't get E = 3.06 as mentiend in the book

That's possible. I also didn't follow your derivation of that equation, as you seemed to come to the conclusion that sin(θ) was zero for reasons I couldn't see. Work with the first two equations and the Q equation.

 

[haruspex]

there is a lot of trigonometric identities but i don't know which one to use :
the one that i used in getting the third equation (sin^2+cos^2=1)
those three equation is correct i checked it :
View attachment 197371

in that case I misunderstood your issue. I thought the book had obtained the third equation from the first two and you were trying to emulate that.

So now it looks to me that your first two equations are momentum equations, and deriving the third is a good move. But now you need to combine this with an energy equation. You are given the net change in energy.