# Graphing a Trajectory with Variable Gravity

1. May 29, 2007

### peterk

1. The problem statement, all variables and given/known data
I would like to graph the position, velocity, and acceleration curves (over time) of an object that is launched straight up from the surface of the Earth, but I need to take into account the fact that the Earth's gravitational pull weakens as the object gets higher.

The position-time graph of an object launched at escape velocity should only go up and never down.

2. Relevant equations
g(r / s)2 = the acceleration of gravity at height s above the center of the Earth (where s > r ).
x(t)=(1/2)*a*t^2+v0*t+x0

3. The attempt at a solution
x(t)=(1/2)*-9.8*t^2+11184*t+0 (assuming that escape velocity is 11184). This would work except there is less gravity as the object moves farther away from the Earth, so it should never really return to the Earth's surface.

2. May 30, 2007

### andrevdh

From

$$a_y = ky^{-2}$$

we progress to

$$\frac{d\ v_y}{dt} = ...$$

and then to

$$\frac{d\ v_y}{dy}\ \frac{d\ y}{dt} = ...$$

which gives

$$v_y\ \frac{d\ v_y}{dy} = ...$$

integrating this gives

$$\int {v_y\ dv_y} = k\int {y^{-2}dy}$$

.....