Solve Repulsive Central Field Integration Problem for Deflection Angle

[Pyroadept]

Homework Statement

Hi, I know this is a mechanics question, but I don't think the actual problem I have with it involves any mechanics, it's just integration techniques.

Find the deflection angle of a particle moving in the following repulsive central field:

U = α/r², α > 0

Homework Equations

Use the formula \int1/(x√(x² - 1))dx = π/2 (π = pi)
where the integral limits are 1 (lower) and ∞ (upper)

The Attempt at a Solution

Hi everyone, here's what I've done so far:

I use the formula χ = | π - 2ϕ_0 |, where χ is the angle of deflection

and then ϕ_0 = \int (ρ/r²√(1 - ρ²/r² - U(r)/E) dr

where the integral limits are r_min (lower) and ∞ (upper)

I am trying to turn this into the form given in the question to apply the formula.

First I factor out a 1/r from inside the square root and sub in the value for U(r):

ϕ_0 = \int (ρ/r√(r² - (ρ² + α/E)) dr


But this is where I get stuck, as I can't see how to turn the (ρ² + α/E) into a 1. Can anyone please point me in the right direction?

Thanks in advance for any help! :)

 

[tiny-tim]

Hi Pyroadept!

(have an integral: ∫ )

Substitute r = [√(r² - (ρ² + α/E))]s

(btw, a simple trig substitution will give you that integral anyway)