Question

imagine an object launched from earth as shown in the image. identify the paths where the object would overcome the gravitational pull of earth.

Explanation:

To overcome Earth's gravitational pull, an object must have a trajectory that takes it far enough away such that it doesn't fall back (escape velocity concept). The paths with the objects moving away from Earth on trajectories that don't curve back towards it (the ones with the arrows pointing away and on paths that are not bound orbits) are the ones where the object overcomes gravity. From the image, the paths of the red dots with arrows moving outward, not following a closed orbit around Earth, are the correct ones. Typically, the paths that are hyperbolic or have enough speed to escape (the ones where the object is moving away and not looping back) are the escape paths. So we identify the paths (the dotted lines with red dots) that are moving away from Earth and not in a bound orbit.

Answer:

The paths of the red - dotted - line trajectories with arrows pointing outward (away from Earth) and not curving back to form a closed orbit around Earth (the ones where the object is moving on a path that will take it beyond Earth's gravitational influence) are the paths where the object overcomes Earth's gravitational pull. (In the image, these are the paths of the red dots with arrows moving in a direction that is not a closed orbit around Earth, e.g., the more "open" dotted - line paths with red dots moving away from the Earth's vicinity.)