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Most importantly, what must change in order for the falling object to change its speed? Is it the distance to the centre of the planet? If you pull the earth away from the object as the object falls, will the object slow down or will it keep accelerating?

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As long as there's a non-zero net force acting on the object, it will have a non-zero acceleration and therefore it will continuously change its velocity: $$\vec{F} = m\vec{a}.$$

In the case of gravity, the force is inversely proportional to the distance between the objects squared: $$\vec{F}_G = G\frac{m_1m_2}{r^2}\frac{\vec{r}}{r},$$ where $\vec{r}$ is the vector connecting the two objects and $r=|\vec{r}|$ its length. So the closer the objects are, the stronger the force or -equivalently- the acceleration. Notice that the acceleration is only zero if the objects are infinitely far apart. (I'm assuming no drag, let's only consider the gravitational force for simplicity)

If you pull the earth in the same direction the object is falling so that you maintain the same distance $r$ at all times the object will just keep falling with a constant acceleration. If you pull it faster, the distance will increase and the acceleration will therefore decrease, meaning the velocity of the object will increase more slowly than before. But it will never decrease. Pulling the earth more slowly will only decrease how much the acceleration would increase if you hadn't pulled, so again the velocity keeps increasing.

So to summarize, the object's velocity will always increase, unless you can get the distance to infinity, which should only take you about - an infinite amount of time. And even then you can only get to a constant velocity, never a decreasing one. You need a repulsive force for that (or additional attractive forces on the other side of the object).

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So if you pulled the earth away at constant velocity, no matter what that velocity is, the object will eventually reach it because its speed would never stop accelerating? – mtanti Mar 30 '13 at 12:12
@mtanti Pulling away the earth at constant velocity actually doesn't make much sense, because a constant velocity implies zero acceleration and therefore zero force, meaning you're not pulling at all. But no, it is possible to keep accelerating the earth away such that the distance between the earth and the object remains the same (I mentioned this in my answer as well) which means by definition that the object will never reach the earth. – Wouter Mar 30 '13 at 12:29
No I mean if the earth wasn't accelerating but moving away at constant velocity, the falling object would always be increasing its speed such that it will eventually reach the earth. – mtanti Mar 30 '13 at 12:32
@mtanti Indeed. Of course our simple picture of newtonian mechanics isn't complete and we need to keep relativity in the back of our minds, but let's say we keep the velocity within the classical range. Getting the earth to relativistic speeds is a herculean task anyway. Note that the earth is already moving, it's not static. So if the object happens to be falling in the same direction as the motion of the earth around the sun (which is close to linear for short time periods) that's the situation you're describing. And for higher constant velocities, the same goes. – Wouter Mar 30 '13 at 12:48
In Newtonian mechanics, an object A with a non-zero acceleration moving in the same direction as an object B with zero acceleration will always catch up with that second object B. You can even calculate when that happens by comparing $v_A(t) = at + v_{0A}$ with $v_B(t) = v_{B0}$. (assuming object A has a constant acceleration this is easy but you can do the same if it is a function of time; and also assuming the motion is linear but again you can generalize this) – Wouter Mar 30 '13 at 12:53

An object accelerates when a force is acting on the object. This given by the Newton's second law $F=ma$, where $F$ is the net force act on the object, $m$ is the mass of the object and $a$ is the acceleration of the object. The reason why objects accelerate as they fall is because the gravity of earth acts on the object. If you pull the earth away from the object as the object falls, the distance between the object and the earth increases. Now the garavitattional varies inversely to the square of the distance.So, the gravitatational force acting on the object due to earth will decrease and therefore the acceleration of the object will also decrease.

what must change in order for the falling object to change its speed

Nothing. As long as there is a net force forcing on the object, the object will accelerate.The acceleration will be given by the Newton's second law.

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But if the earth were the be pulled away from the falling object, could the object return to a previous speed with regards to falling? Or will its speed never slow down? – mtanti Mar 30 '13 at 10:15
There will always be a net accelerating force. So even if you pull the earth away, there will be a force, and thus acceleration. – Bernhard Mar 30 '13 at 12:04
So if you pulled the earth away at constant velocity, no matter what that velocity is, the object will eventually reach it because its speed would never stop accelerating? – mtanti Mar 30 '13 at 12:25

I think this may be also an possibility for explaining your answer.. This is also an attempt though the real answer has already been explained. Whenever you throw an object up, it always tries to fall down because it is attracted by gravitational force. Now suppose you throw an object to height h then, it will gain P.E as mgh and K.E 0.But since it is attracted by gravitational force it will fall down.. Let it reach at height x from the ground surface. Now its new *PE * is mgx and KE is somehow more than previous.. ( from the conservation of energy).. Since it has gain some new KE that is more than previous one , it will directly mean that it has gain some velocity.. As there is difference in velocity, there will be of course an acceleration produced. So, in any case if any object is falling toward mass.. then object will always accelerate , provided that Gravitational force is the only force acting.

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Most importantly, what must change in order for the falling object to change its speed?

Obviously, there's nothing necessary for that. Ignoring drag, the velocity of the object is increasing for sure. Only the acceleration remains constant. Taking air effects into account, the object attains the terminal velocity (based on fluid), its $v$ remains constant & $a$ becomes zero. If you're concerned about changing its velocity now, then you either apply another force or remove the fluid...

Is it the distance to the center of Earth?

Well, quite... Once the distance to the center increases, the gravitational pull exerted on the object decreases - leading to less acceleration. Hence, velocity changes indeed...

If you pull the earth away...

The acceleration on the object decreases slowly. If another body attracts it with some comparable force, your object stops accelerating towards it. But, never quite reaches stationary state...

Whenever a force is exerted on an object, there's a change in its velocity as a function of time. It's old-school. Check out Newton's laws and finally substitute gravity and you're done...

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Are you sure about the deceleration? There is still a net accelerating force, right? – Bernhard Mar 30 '13 at 12:06
@Bernhard: Oops.. A small mistake (I just sped up) - Corrected now. Okay you're right :D – Waffle's Crazy Peanut Mar 30 '13 at 12:12
How do you decelerate the object? Wouldn't you need a negative force for that? – mtanti Mar 30 '13 at 12:13
@CrazyBuddy I don't know how you would go about moving the earth around, but that is a completely different question :) – Bernhard Mar 30 '13 at 12:14
@mtanti: Yep, Of course. You can see that in the vector equation $\vec{F}=m\vec{a}$ ;-) – Waffle's Crazy Peanut Mar 30 '13 at 12:17

protected by Qmechanic May 12 '13 at 17:23

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