arildno said:
... Two spheres, each with a mass of 1 kg and a radius of 1 meter, lie in space. Their centers are 10 meters apart. When will they make contact? ... arildno's posts from that old thread with latex cleaned up:
Call the spheres A and B, and let the origin of the coordinate system lie at the midpoint of the line segment defined by their centres.
Calling the sphere centres' positions as functions of time
x_{A} (t) , x_{B} (t) , x_{A} (0) = -5 , x_{B} (0) = 5
respectively, we define the distance function between them as:
D(t)=x_{B} (t) - x_{A} (t) , D(0) = D_{0} = 10
Setting up Newton's 2nd law for both, we get, with unit masses:
\frac{d^{2}x_{A}}{dt^{2}} = \frac{G}{D^{2}}
\frac{d^{2}x_{B}}{dt^{2}} = -\frac{G}{D^{2}}
whereby the equation for d(t) is readily derived:
\frac{d^{2} D}{dt^{2}} = -\frac{2G}{D^{2}} (*)
We assume that the initial velocities are 0, i.e
\frac{dD}{dt}\mid_{t=0}=0
Let us multiply (*) with dD/dt:
\frac{d^{2}D}{dt^{2}}\frac{dD}{dt}=-\frac{2G}{D^{2}}\frac{dD}{dt}
Integrating both sides from t=0 to some arbitrary t-value, taking due notice of the initial conditions, yields:
\frac{1}{2}(\frac{dD}{dt})^{2}=\frac{2G}{D}-\frac{2G}{D_{0}}
multiplying with two, taking the square root and remembering that D(t) will be decreasing, we get the diff. eq:
\frac{dD}{dt} = -\sqrt{ \frac{4G}{D_{0}}} \sqrt{ \frac{D_{0} - D} {D}}
This is a separable diff.eq; we write:
\sqrt{ \frac{D}{D_{0} - D}} dD = -\sqrt{ \frac{4G}{D_{0}}} dt
We now remember that when they spheres make contact, D(T)=2, where T is the time we're looking for! Thus, we get the equation for T, integrating both sides:
\int_{10}^{2} \sqrt{ \frac{D}{D_{0}-D}} dD = -\sqrt{ \frac{4G}{D_{0}}} T
or equivalently:
T = \sqrt{ \frac{D_{0}}{4G}} \int_{2}^{10}<br />
\sqrt{ \frac{D}{D_{0}-D}}dD
In order to crack that integral, let us set:
u=\sqrt{\frac{D}{D_{0}-D}}\to{D}=D_{0}-\frac{D_{0}}{1+u^{2}}
dD=\frac{D_{0}2u}{(1+u^{2})^{2}}du
The limits are
D=10\to{u}=\infty,D=2\to{u}=\frac{1}{2}
We thereby get the expression for T in u:
T = D_{0} \sqrt{ \frac{D_{0}}{G}} \int_{ \frac{1}{2}}^{ \infty} <br />
\frac{u^{2}du} {(1+u^{2})^{2}}
doing integration by parts.
\int \frac{1}{1+u^{2}}du = \frac{u}{1+u^{2}} +<br />
\int \frac{2u^{2}}{(1+u^{2})^{2}}du
\int \frac{u^{2}}{(1+u^{2})^{2}}du = <br />
\frac{1}{2}( \arctan(u) - \frac{u}{1+u^{2}})+C
and
T = \frac{10^{ \frac{3}{2}}} {2 \sqrt{G}}<br />
( \frac{ \pi}{2} - \arctan( \frac{1}{2}) + \frac{2}{5})