Answer:
The unknown quantities are:
E and F
The final velocity of the proton is:
√(8/3) k e^2/(m*r)
Explanation:
Hello!
We can solve this problem using conservation of energy and momentum.
Since both particles are at rest at the beginning, the initial energy and momentum are:
Ei = k (q1q2)/r
pi = 0
where k is the coulomb constant (= 8.987×10⁹ N·m²/C²)
and q1 = e and q2 = 2e
When the distance between the particles doubles, the energy and momentum are:
Ef = k (q1q2)/2r + (1/2)m1v1^2 + (1/2)m2v2^2
pf = m1v1 + m2v2
with m1 = m, m2 = 4m, v1=vf_p, v2 = vf_alpha
The conservation momentum states that:
pi = pf
Therefore:
m1v1 + m2v2 = 0
That is:
v2 = (1/4) v1
The conservation of energy states that:
Ei = Ef
Therefore:
k (q1q2)/r = k (q1q2)/2r + (1/2)m1v1^2 + (1/2)m2v2^2
Replacing
m1 = m, m2 = 4m, q1 = e, q2 = 2e
and v2 = (1/4)v1
We get:
(1/2)mv1^2 = k e^2/r + (1/2)4m(v1/4)^2 = k e^2/r + (1/8)mv1^2
(3/8) mv1^2 = k e^2/r
v1^2 = (8/3) k e^2/(m*r)
