Laws & Equations of Vertical Motion

Before Galileo, scholars believed that if a heavy and a light object are dropped from the same height, the heavier object will reach the ground first. The reason for the rise of this idea was certainly everyday experiences or common observations. Galileo in 1589 dropped various heavy objects from the Leaning Tower of Pisa in Italy and showed that the objects reached the ground almost simultaneously. According to him, the slight difference in time seen is due to air resistance. Later, Sir Isaac Newton proved the truth of Galileo’s theory.

Galileo established several laws for freely falling objects (ignoring air resistance). These laws are known as the laws of falling objects.

Laws of falling bodies:

The Earth attracts all objects near it towards its center. This force of attraction is called Gravity. We see that if an object is allowed to fall, it falls straight down under the influence of gravity. This fall of an object under the action of gravity alone is called free fall. The acceleration with which a body falls freely is equal to the acceleration due to gravity.

First Laws of Falling Bodies:

All objects fall at the same speed during free fall from a height.

According to this law, if there is no air resistance, all objects, whether light or heavy, will fall at the same speed at the same time. And if there is air resistance, the lighter object will touch the ground later than the heavier object. Galileo could not prove this formula experimentally because the air pump was not invented at the time. After the invention of the air pump, Newton proved the truth of this formula with his famous guinea and feather experiment.

The experiment of Guine & Feather:

In this experiment, a guinea (i.e. a heavy object) and a feather (i.e. a light object) are placed in a glass tube with a diameter of about 5 cm and a gap between the two ends of the tube is one meter long.

If the tube is suddenly turned upside down when it is full of air, it is seen that the guinea falls down faster than the feather. Now if the tube is emptied with the help of a pump and the tube is turned upside down again, it is seen that the guinea and the feather reach the other end of the tube at the same time.

This experiment proves that, in the absence of air resistance, all objects, heavy or light, fall at the same speed in the same instant. Therefore, all falling objects at the same place experience the same acceleration.

Second Laws of Falling Bodies:

The velocity acquired by the object falling freely from a height at any time is proportional to the time of fall.

According to this law, the velocity of a freely falling object from rest increases gradually with time. If the object falls for some time (t) and if the velocity acquired by the object at the end of that period is v, then,

v ∝ t

or, v/t = constant

That is, the rate of change of velocity or acceleration is constant. Therefore, it can be understood from the second formula that the acceleration of an object during free fall under the action of the reaction always remains constant.

Info: Objects in free fall experience constant acceleration due to gravity (g = 9.8 m/s²)

Third Laws of Falling Bodies:

The distance covered by the object of a freely falling body at any time is proportional to the square of the time of fall.

Suppose, a body starts falling from rest and falls freely to a height h at time t. Then according to the third formula

h ∝ t²

or, h/t² = constant

This means that if the object travels a distance x in the first second during free fall, it will travel a distance of 4x and 9x in the second and third seconds respectively.

Info: When an object moves upward, its velocity decreases by g every second and when it moves downward, its velocity increases by the same amount i.e. g every second.

Equations of Vertical Motion under Gravity

We have seen that the downward motion of an object under the action of gravity is a uniformly accelerated rectilinear motion. Therefore, the equations of ordinary rectilinear motion are applicable in this case as well. The equations of rectilinear motion with uniform acceleration are –

1. v = u + at
2. s = ut + (1/2)at²
3. v² = u² + 2as.

Where-

• s = displacement
• u = initial velocity
• v = final velocity
• a = acceleration
• t = time.

For a falling body downward

In the case of vertical motion, we used the height h instead of displacement s, and gravitational acceleration g instead of normal linear acceleration a. So the equations of vertical motion under gravity now become –

1. v = u + gt
2. h = ut + (1/2)gt²
3. v² = u² + 2gh.

For a falling body if the initial velocity u is zero i.e. the body falls from rest from a certain height then,

1. v = gt
2. h = (1/2)gt²
3. v² = 2gh.

For a body moving upward

When an object is thrown vertically upwards, the velocity of the object gradually decreases. At one time, the velocity is zero for a moment. Therefore, the motion of the object thrown vertically upwards is a decelerating motion. If the upward direction is taken as positive, the equations of motion of the object are-

1. v = u − gt
2. h = ut − (1/2)gt²
3. v² = u² − 2gh.

Equation of Maximum height

The object is thrown vertically upwards with velocity u. If the object rises to a maximum height H, then it can be said that when h = H then the final velocity at maximum height v = 0

0 = u² − 2gH

or, H = u²/(2g)

Equation of Time of Ascent

If the object rises to a maximum height at time t = T, and the final velocity at maximum height v = 0

0 = u − gT

Or, T = u/g

As we know when an object is thrown vertically upwards, the velocity of the object gradually decreases and at a certain time, the velocity becomes zero for a moment. After that the object starts falling downwards with increasing velocity and just before touching the ground, the velocity of the object is equal to its initial velocity. It is seen that the period of rise and fall of the object is exactly equal. From this it is understood that the upward and downward motion of the object are similar in all respects but opposite in direction.

Therefore if T_up is the time taken to reach the maximum height and T_down is the time taken to reach the ground then, T_up = T_down.

So the total time of Flight is T’ = T_up + T_down = 2T

T’ = 2u/g

Info: If air resistance is ignored, the ascent time = descent time for an object thrown vertically upward.