This may belong in the physics stack exchange, but please hear me out and it does involve mathematics. Two planets, both of mass m, are separated by a distance x (that is from one planet's center to the other's) and the only force that acts upon them is gravity. How long do they take to collide? (Solve symbolically) The acceleration is not constant, because the planets get closer every second, and increases as jerk (the derivative of acceleration). I smell an integral somewhere because finding time involves adding up all the little distances dx passed by a planet as a little dt passes. And since they are the same mass, the distance traveled by one will be the same as the other, and will be half x. Also, I need a velocity function to integrate, and since velocity * time is distance, I set up int(v(t)dt)=x/2 and solved for v(t), but the problem is I got v(t)=(xt)/4, which doesn't make sense because I was expecting a quadratic because there is not constant acceleration.
The formula for acceleration is equal to (Gm^2)/(x^2) where G=6.67408e-11 (Nm^2)/(kg^2) but the problem is that acceleration is not constant. One could try to take the derivative of that function with respect to x, as acceleration will increase linearly with every dx.
What is the correct way to solve this? Thanks.
Take the line joining the planets as the x axis and the midpoint between them as the origin. Assuming at t = 0 v = 0 and x = D (initial distance between the planets is 2D) then we have
$$-\frac{Gm^2}{(2x)^2} = m\frac{dv}{dt}$$
$$-\frac{Gm}{4x^2} = \frac{dv}{dx}\frac{dx}{dt} =\frac{vdv}{dx}$$
$$-Gm\frac{dx}{x^2} = 4vdv$$
$$\frac{Gm}{x} = 2v^2 + const$$
$$at t = 0 v = 0 and x = D we have Const = \frac{Gm}{D}$$
$$\frac{Gm}{x} = 2v^2 + \frac{Gm}{D}$$
$$\sqrt{\frac{Dx}{D-x}}dx = -\sqrt{\frac{Gm}{2}}dt$$
Integrating both sides we have
$$-\sqrt{D}\sqrt{x(D - x)} + D\sqrt{D}(tan^{-1}\sqrt{\frac{x}{D - x}} - \frac{\pi}{2}) = - \sqrt{\frac{Gm}{2}}t$$
This is the equation in x and t. Let's take D as the 1\2 of the distance between the sun and the earth as 74.8 x $10^9 m$
Mass of the earth as 5.972 x $10^{24}$ kg
And x = R at collision R is the radius of earth as 6,371,000 m
The planets will hit each other after 72 years.