Neutron collides elastically with a helium nucleus

[amazondog]

"neutron collides elastically with a helium nucleus"

Homework Statement

A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times that of the neutron. The helium nucleus is observed to rebound at an angle θ'2 = 40° from the neutron's initial direction. The neutron's initial speed is 5.0 e5 m/s. Determine the angle at which the neutron rebounds, θ'1, measured from its initial direction.

Homework Equations

(1/2)mu1^2 = (1/2)(4m)(v2^2) + (1/2)(m)(v1)^2 where u1 = 5.0e5m/s
(1/2)u1^2 = (1/2)(4)(v2^2) + (1/2)(v1)^2
u1^2 = (4)(v2^2) + (v1)^2

m(u1)(cos40) = (4)(m)(v2) + (m)(v1x)
(u1)(cos40) = (4)(v2) + (v1x)
(u1)(.7660) = (4)(v2) + (v1x)
(5.0e5)(.7660)
383022.2216 = (4)(v2) + (v1x)
v1x = 383022.2216 - (4)(v2)

m(u1)(sin40) = (m)(v1y)
(u1)(sin40) = (v1y)
(5.0e5)(.6428) = (v1y)
v1y = 321393.8048

(v1^2) = (v1x)^2 + (v1y)^2
(v1^2) = { (383022.2216 - (4)(v2) }^2 + ( 321393.8048)^2
(v1^2) = (1.4671e11 - 16v2^2 + 1.0329e11)
(v1^2) = 2.50e11 - 16v2^2
(v1) = 5.0e5 - 4v2)

u1^2 = (4)(v2^2) + (v1)^2
(2.50e11) = (4)(v2^2) + (2.5e11+16v2^2)
5.0e5 = 4v2 + 5.0e5+4v2
0 = 8v2 NOOOO!


Thanks, any help is appreciated!

 

[ehild]

Show a drawing, please. I do not see which is the 40° angle, and what is u and what is v.

ehild.

 

[lightgrav]

you seem to be using u for initial velocities, v for final velocities.
interesting to have used alpha's final direction as "x" .
The last step in set 4 is bogus ...

 

[amazondog]

Should it be v1 = 5.05e5 + 16v2?

(v1^2) = (v1x)^2 + (v1y)^2
(v1^2) = { (383022.2216 - (4)(v2) }^2 + ( 321393.8048)^2
(v1^2) = (1.4671e11 - 16v2^2 + 1.0329e11)
(v1^2) = 2.50e11 - 16v2^2
(v1) = 5.0e5 - 4v2)

 

[lightgrav]

well, let's see ... I remember that 3^2 = 5^2 - 4^4 ...
so you think that implies 3 = 5 - 4 ?
you do the same thing again in set 5 ...

If it were me, I would only analyze collisions in the center-of-momentum frame .