The Impossible Task of Solving a 2nd Order DE for r(t)

[Repainted]

Suppose an object is moving toward the Earth(with a direction perpendicular to the Earth's surface) at an initial speed v₀, starting from a distance r₀. The object also experiences gravitational acceleration. Is it possible to obtain an expression of r as a function of t? In other words, what's the distance from the Earth at a time t?

I tried solving this second order differential equation:
d²r/dt² = GM/r²

I ended up with a mess of stuff. And it seems impossible to make r the subject of the equation.

 

[Valerie Park]

The answer is yes, it is possible to obtain an expression of r as a function of t. To do this, you can solve the differential equation d2r/dt2 = GM/r2 with the initial conditions r(0) = r0 and v(0) = v0, where G is the gravitational constant and M is the mass of the Earth. The solution is given by r(t) = (GM/v02) + r0cos(v0t/r0) - [(GM/v02) + r0]cos(v0√(2GM/r0)/v0t).

 

[sir-pinski]

Unfortunately, it is not possible to obtain an explicit expression for r(t) in this scenario. This is because the equation is a second order differential equation, which means it contains a second derivative of r with respect to t. This makes the equation more complicated and difficult to solve for r as a function of t.

Additionally, the equation contains the gravitational constant G, the object's initial speed v0, and the object's initial distance from Earth r0. These variables cannot be eliminated from the equation, making it impossible to obtain a simple expression for r(t).

However, it is still possible to obtain a numerical solution for r(t) by using numerical methods such as Euler's method or Runge-Kutta method. These methods involve breaking down the equation into smaller steps and using iterative calculations to approximate the value of r at different time intervals. This will give an approximate value for r at different points in time, but it will not provide a simple expression for r(t).

In conclusion, while it is not possible to obtain an explicit expression for r(t) in this scenario, numerical methods can still be used to approximate its value at different points in time.