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Is there any force on the object being applied in that case? I think there is not since acceleration is zero. So does that mean no force is also being applied back on my thumb?

Also, when the object was rest. And at that instant, when I applied the force on object to impart a certain velocity in it, it should have applied a force back on my thumb in the opposite direction. So the acceleration on my thumb should have been in opposite direction to the direction of the force applied on the object. So, my thumb should have moved in opposite direction, but why does it move in the direction of the object? I am really confused.

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Following proton's answer, one part of your question can be answered as follows. In space, an object will move at constant velocity (possibly zero) without any force applied (from your thumb for example). So you could give it a small nudge, and then it would continue with constant velocity without you applying any further force. On the other hand, when the object is sitting on a surface on Earth, then when the object is moving across the surface, that surface provides a force (friction) in the opposite direction to the motion, so that its velocity is reduced and generally quickly brought to zero. Hence in order to maintain a constant velocity for the object, you will have to be constantly providing a force, by continuing to push with your thumb. In terms of Newton's second law (as applied to the forces acting on the object), the acceleration can be zero because we have $m_{\text{object}}a_{\text{object}}=F_{\text{thumb}}-F_{\text{friction}}$ and these two forces can cancel if you push with the right force.

On the other part of your question, it is true that if you apply a force to the object (through your thumb for example), you will experience an equal and opposite force. This is what you feel on your thumb when you push the object. In space, this would mean that when you push the object slightly, the object will be sent forward and you will also be sent floating backwards (at a slower speed, but with exactly opposite momentum as the object). On Earth, the situation is more complicated. When you push something, you usually keep your feet firmly planted on the ground (the ground is providing a frictional force great enough to stop you from moving, or perhaps you have some foothold etc). So while you are experiencing the equal and opposite force, you transfer the change in momentum to the Earth itself (you feel a force both at your thumb and at your feet), and so ultimately the Earth picks up an (imperceptibly small) change to its rotation.

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Of course there is force on the object. Since you are not in space, friction is acting on your object, so if it's moving in constant velocity it means the sum of forces on it is zero, so you must be acting force on it as well.

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Your thumb is elastic; if it presses against anything, you know there is a force because you can feel it (your thumb senses the pressure). Whether the object is in motion or not, if your thumb isn't glued to it, the fact that your thumb remains in contact proves that there is a force (pressure) exerted, otherwise the object would drift out of contact.

Acceleration of an object is due to the sum of ALL forces on that object.

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