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For example, if I threw a 1 kg stone and a 100 kg stone vertically upward with the same velocity will they reach the same height, if yes then how?

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    $\begingroup$ A closely related question points out that the motion of the Earth is different in response to larger falling masses, though that motion is negligible for both of the examples in your question. $\endgroup$
    – rob
    Commented Aug 9, 2020 at 8:48

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Yes.

The key aspect in your scenario is: they have the same initial velocity. In your addendum „if yes then how“ you seem to overlook that it takes much more force to accelerate the 100kg stone to same velocity. But when they have the same velocity after you let loose they follow the same trajectory, meaning they reach the same height.

There is, however, one aspect to keep in mind: the masses of both stones have to be much smaller than the mass of earth. Otherwise their masses will begin to influence the movement of earth itself. Btw. Stones that big are called Planets or moons 😉

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The answer is yes, they reach the same maximum height, provided we include only the gravitational force (no air effects).

The reason can be seen in different ways.

  1. Equations of motion. Acceleration in a constant gravitational field is independent on the mass ($m a = F$, but $F$ is proportional to $m$). And only acceleration rules the maximum height.
  2. Conservation of energy. The maximum height in a vertical throw is attained when the whole initial kinetic energy ($\frac12 m v_0^2$) has been transformed into potential energy ($mgh$). Once again, the same mass factor appears in both expressions and it cancels.
  3. Dimensional analysis. If one tries to combine the possible relevant data (initial velocity $v_0$, gravitational acceleration $g$, and mass $m$) to get a quantity of physical dimension of length ($m^{\alpha}g^{\beta}v_0^{\gamma}$), the mass should appear with power $\alpha=0$. I.e. it cannot play a role.
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imagine you and 100 other people each have a 1kg stone. (assume you can all throw your stone at the same velocity). You throw your stone up and it reaches a certain height. Then the other 100 people throw their stone up at the same time. What happens ? 100kg worth of stone reaches the same height as your 1kg worth of stone.

Just because that separate 100kg worth of stones are not connected by a chemical bond does not affect the height they reach. You can think of the stones being connected by massless rods if that helps you think of the individual 100 stones as a connected whole 100kg mass. That's basically what a chemical bond does it connects the mass without adding mass. You can see it requires 100 times the force for the 100kg.

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