Calculating how long it takes for a comet to reach a sun of radius zero

[xdrgnh]

Homework Statement

Consider the extreme case that the comet is released from rest at a distance R_max from the sun. In this case L is actually zero. Use the technique described in connection with 4.58 to find how long the comet takes to reach the sun. The radius of the sun for now is zero.

Homework Equations

Equation 4.58 is T=∫dr/(E-U(r))^(1/2)

E should be -GM_sM_c/R_max

and U(r) should be -GM_sM_c/r

The Attempt at a Solution

So I intergrate this and my bound are 0 to R_max. I just want to know if I'm on the right track becasue the integral is very messy. Thanks.

 

[vela]

Is T supposed to be the time? If so, equation 4.58 doesn't seem to work out unit-wise.

 

[xdrgnh]

Yes it is. Well I forgot to mult that integral by a (m/2)^(1/2). But that doesn't change the difficulty of the intergrand.

 

[vela]

Without that factor, the units of T are 1/force. Multiplying by mass^1/2 won't fix it.

In any case, I don't see why you think the integrand is a mess. It looks like it should succumb to a simple u-substitution.

 

[xdrgnh]

Darn I made another mistake while typing I'll fix the OP re look at it. I added a square root.

 

[cepheid]

$$ v = \frac{dr}{dt} \Rightarrow dt = dr/v $$ $$v = \sqrt{2K/m} = \sqrt{2[E - U(r)]/m}$$ $$t = \int\,dt = \int v^{-1} dr = \int \left(2\frac{E - U(r)}{m}\right)^{-1/2}\,dr$$

Is that the equation?

 

[xdrgnh]

Yep.

 

[xdrgnh]

Any word if my E-U(r) is correct?

 

[cepheid]

Any word if my E-U(r) is correct?

Yeah, the expression $$- G\frac{mM}{R} + G\frac{mM}{r} $$ looks okay to me. So the integrand would become $$\left[2GM \left(\frac{1}{r} - \frac{1}{R}\right) \right]^{-1/2} $$ Did you have any luck computing it?

 

[xdrgnh]

Not much luck. I haven't integrated any complicated integral since I took calc II two year ago. But I feel a trig sub might work. But I was mainly concerned with the physics part rather then the math. I doubt a question like this would appear on the exam and I',m only doing it for my own enrichment. But see how the trig goes if I can figure out how it would work. If not I might try a u sub. I know I should simplify it first.