Thinking about Free Fall

Two spheres, M and N of mass 6kg and 4kg respectively are in free fall.
Which statement/s is/are true with regards to the masses?

I- The velocities of M and N are the same.
II- The momentum of M and N are the same.
III- The gravitational force of M and N are the same.
IV- The acceleration of M and N are the same.
V- Kinetic energy of the two masses are the same.

Let's do some thinking according to what we have learnt previously in chapter 2.

What will be your answer?

VELOCITIES

We know that

Velocity = Displacement over Time or rate of change of the position of an object.

Speed or velocity increase with time. 

v=u + at

where v, final velocity
u, initial velocity
a, acceleration
t, time

Logically, if we assume the objects free fall,

u=0 for both, then, the velocity depends on acceleration (a) and time(t).
 
But, if we consider a more complex situation where there is air resistance acting on the objects that have been dropped, the velocities of the falling masses are not equal. Each falling object will eventually reach a terminal velocity. The terminal velocity depends on many factors including mass, drag coefficient, and relative surface area and will only be achieved if the fall is from sufficient altitude. A free falling object achieves its terminal velocity when the downward force of gravity (Fg) equals the upward force of drag (Fd).

As formula for terminal velocity include mass as a factor, then the velocities of the masses are not the same.

MOMENTUM

For momentum of M or N

p = mv
momentum for M will depend on its mass and velocity, so is momentum for N.

Momentum depends on mass and velocity. As the masses are different, the momentum of the masses are therefore not the same.

GRAVITATIONAL FORCE

In free fall, object is subjected to only one external force, the gravitational force, which is also the weight of the object. In other word, for a free falling object, the net external force is just the weight of the object.


F-W = 0
F=W

And Weight = mass * gravitational acceleration

We know that the weight equation is product of the mass of the object and the gravitational acceleration, g     ( see more in ACCELERATION). Gravitational acceleration, g decreases with the square of the distance from the centre of earth. However, in most cases, and for practical reasons, we assume the factor to be constant. This value, g is 9.8 m/s^2 on earth surface.

In reality, in atmosphere, there are other forces acting on the object such as air resistance or drag force.

The value of gravitational force for each mass will depend on the masses of the objects.

ACCELERATIONS

In free fall, the acceleration of both objects equals the gravitational acceleration. Don't be fooled, This is not the same as gravitational force. Gravitational acceleration unit is m/s/s. The symbol g stands for the acceleration of gravity - a quantity whose value on Earth's surface is 9.8 m/s/s.


Gravitational force = Mass * gravitational acceleration

Or

W = m * g

Using Newton's second law, Force = mass * acceleration

In free fall, the only force acting on the object will be the gravity : 

F = W

From the Newton Second Law above,
F= m *a  and W = m*g

Hence,  m * a = m * g

Therefore, a= g

In free fall, masses of the objects are irrelevant.

As an elaboration, remember in physics class we learn that acceleration of an object is directly proportional to force and inversely proportional to mass. 

Therefore, if we increased force, we increased acceleration but if we increased mass we tend to decrease acceleration. Hence, the greater force on more massive objects is offset by the inverse influence of greater mass. In other words, acceleration is the ratio of Force over Mass.

Subsequently, all objects free fall at the same rate of acceleration, irrelevant of their mass. 

Kind of hard to imagine that an elephant will fall at the same time as a rat?
And it was told, Galileo, the Italian scientist, dropped two balls of different masses from the Leaning Tower of Pisa and demonstrated their time of descent was independent of their masses.

KINETIC ENERGY

The formula used to find how much the kinetic energy for each mass will be by using the formula:

K.E = 1/2 x mass x velocity^2

If the objects are dropped from height h = 3 m, then the velocity just before impact is
v = 7.668 m/s.

If the mass is m = 6 kg, then the kinetic energy just before impact is equal to
K.E. = 176.4  J, which is of course equal to its initial potential energy. For m= 4 kg, the K.E = 117.6 J

The accuracy of this calculation depends upon the assumption that air friction is negligible, and that the height of drop is small compared to the radius of the earth.


Credits:
 
free fall

Free fall and air resistance  

Terminal velocity