Graphing a Trajectory with Variable Gravity

[peterk]

Homework Statement

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.

Homework 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

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.

 

[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

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

...

 

[m2003]

I would first recommend using appropriate units for the equations, such as meters for distance and meters per second squared for acceleration. Additionally, it is important to consider the direction of the object's motion, as the gravitational force will act in the opposite direction of the object's motion as it moves away from the Earth.

To accurately graph the trajectory of an object with variable gravity, we can use the equations of motion for projectile motion, taking into account the changing acceleration due to gravity. This would involve breaking down the object's motion into smaller segments, where the acceleration due to gravity changes at different heights. This can be done by using the equation g = GM/r^2, where G is the gravitational constant, M is the mass of the Earth, and r is the distance between the object and the center of the Earth.

We can also use numerical methods or computer simulations to accurately model the trajectory of the object, taking into account the changing acceleration due to gravity. This would provide a more precise graph of the object's position, velocity, and acceleration over time.

In terms of the position-time graph, it is correct that the object should never return to the Earth's surface if it is launched at escape velocity. This is because at that velocity, the object has enough kinetic energy to overcome the gravitational potential energy and escape the Earth's gravitational pull. However, the object will continue to move away from the Earth, with its velocity decreasing as it moves farther away. This can be seen in the velocity-time and acceleration-time graphs, where the velocity and acceleration decrease as the object moves away from the Earth.

In conclusion, graphing the trajectory of an object with variable gravity requires careful consideration and the use of appropriate equations and methods. By taking into account the changing acceleration due to gravity, we can accurately model and graph the object's motion over time.