Will the acceleration of the object and the acceleration due to gravity cancel each other and that object remains at the same space coordinate?

  • $\begingroup$ F=ma, but you should calculate sum of all forces first! And only then calculate the acceleration by the sum of forces. In your example the sum of forces is zero. $\endgroup$
    – Heopps
    Sep 29 '18 at 7:28

This might be a trick on words. If an object sits on a table, there is a gravitational force pulling it down, and there is an equal and opposite upwards force from the table. So we say that the forces cancel because $F_{Gravity} + F_{Table}=0$. We don't say that the table accelerates it upwards and cancels the gravitational acceleration downwards.

If the direction $\mathbf{\hat{r}}$ points upwards, then on the surface of the Earth with radius $r_E$:

$$F_{Gravity} = -\frac{G \ m_E \ m_{obj}}{r_E^2} \ \mathbf{\hat{r}}$$

But maybe we want to talk about the Earth's gravity without talking about any particular object. We could write the gravitational force per unit mass as

$$\frac{F_{Gravity}}{m_{obj}} = -\frac{G \ m_E}{r_E^2} \ \mathbf{\hat{r}}$$

That term on the right is called "gravitational acceleration" and it's a way to express how strong gravity is at Earth's surface.

$$\frac{G \ m_E}{r_E^2} = g$$

and on Earth's surface $g$ is about 9.81 m/s².

So to the original question:

Will the acceleration of the object and the acceleration due to gravity cancel each other and that object remains at the same space coordinate?

The forces cancel, but the net acceleration of an object that continues to remaining at the same space coordinate is just zero.

While we express the strength of the gravitational field in units of acceleration, and call that number gravitational acceleration, the object is not accelerating and you don't need to do additional acceleration in the opposite direction to keep it in one place.

  • $\begingroup$ So will the object ever escape the earths gravity if it maintains its acceleration? $\endgroup$
    – Poin
    Sep 29 '18 at 6:29
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    $\begingroup$ @Dhruva If something maintains its acceleration, yes, that's a mathematical certainty, but physically and realistically, things don't ever maintain acceleration. I think that this is getting too hypothetical for a practical discussion, and off-topic for this particular Stack Exchange site about Space Exploration. I think you can check out other SE sites as well, like Physics SE but I'd advise you to look for similar questions first before asking a new one. $\endgroup$
    – uhoh
    Sep 29 '18 at 6:44
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    $\begingroup$ Seems to me any discussion of constant acceleration on SpEx would be incomplete without a mention of powered hover.... $\endgroup$ Sep 29 '18 at 7:19
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    $\begingroup$ uhoh is correct twice. Acceleration is simply a property of an object itself, not of the individual forces acting on it. And this question does belong on Physics.SE. $\endgroup$
    – DrSheldon
    Sep 29 '18 at 12:10
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    $\begingroup$ uhoh is thirdly correct: there are similar questions on Physics.SE. @Dhruva, please look at physics.stackexchange.com/q/160538/200019 and physics.stackexchange.com/q/336413/200019. Three times in one day... uhoh's clock must be experiencing time dilation! $\endgroup$
    – DrSheldon
    Sep 29 '18 at 16:15

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