Why bodies with different masses reach gound at the same time in vacuum?

Question

According to Newton's Gravitation Equation $$F=G \dfrac{M\cdot m}{r^2}$$ force of gravity depends upon mass of both objects, and $$F=ma$$ means force is proportional to acceleration. Then why do objects with different masses reach the ground at the same time in vacuum?

Answer 1

The time it takes to reach the ground is determined by the acceleration of the object. If both objects accelerate in exactly the same way, then both will reach the ground at the same time.

We can easily derive the acceleration in this case, which is:

$$a=\frac{F}{m}=\frac{\frac{GMm}{r^2}}{m}=\frac{GM}{r^2}$$

As you can see, the mass $m$ of the object completely cancels out, so the acceleration depends only on the mass of the Earth (or other gravitating planet) and the distance from the center. The object's mass simply doesn't matter.

Answer 2

Gravitational force acts separately on all the individual particles in the body which means that all the particles will be accelerated by the force of gravity in the same manner irrespective of the body's mass.

Let's say you have a body made up of two particles having the same mass ($1$ u) and hence during free fall both the particles will experience the same force due to gravity given by the Gravitation equation. Hence as a whole, the body experiences twice the force experienced by a single particle. Now when we try to determine the acceleration we notice that $$a=\dfrac{2\cdot F}{2\cdot u}\text ms^{-2}$$

Now if we have $n$ such particles, all the $n$ particles separately will experience the same force given by the Gravitation equation and hence again, as a whole, the body experiences $n$ times the force experienced by a single particle. Hence the acceleration of the body will be $$a=\dfrac{n\cdot F}{n\cdot u}\text ms^{-2}$$

In both the cases we see that though the individual values of the force applied and the total mass differ with cases the ratio of the the two always remains the same.

Hence all objects accelerate uniformly with a value of $9.8 \text ms^{-2}$.

Therefore in both cases the acceleration of the center of mass of the body is the same.

Answer 3

A simple answer, it reaches the ground on the same time, because their acceleration is the same, which is -9.8m/s^2, and also in vacuum there have no air resistance, so they should hit the ground the same time.