Table of Contents
GRAVITY AND ACCELERATION DUE TO GRAVITY
In this universe each body attracts every other body. The attraction between different planets is known as gravitation. The earth we live in is also a planet. The attraction exerted by this planet on other bodies towards its centre is known as gravity.
So, gravity is a force and it is due to this gravity that all bodies on and around the earth are attracted towards its centre. It is due to this gravity that when a ball is thrown upwards, it comes back to the earth or when a particle of stone is let fall from certain height, it falls to the earth and does not stay in the space at which it was released.
It is from our practical experience that when a body falls down, its velocity gradually increases. From this phenomenon it can be concluded that the falling bodies are subjected to acceleration. It is from Newton’s second law of motion that when a force acts on a body, the body moves with some acceleration. Since during falling, a body is under the action of gravitational force of the earth, it is quite likely that it will move with some acceleration. This acceleration is called “acceleration due to gravity” and is always denoted by ‘g’.
The value of the acceleration due to gravity is 9.807 m/s² which is invariably taken as equal to 9.81 m/s².
POSITIVE AND NEGATIVE VALUE OF 'G'
When a body falls vertically downward under the action of gravity, its motion is accelerated and the acceleration due to gravity which acts in the same direction as the direction of motion of the body, is taken as positive.
When a body is thrown vertically upward, it’s motion is retarded, because in this case the acceleration due to gravity act in a direction opposite to the direction of motion of the body. In this case, the acceleration due to gravity is taken as negative acceleration.
EQUATIONS OF MOTION UNDER GRAVITY
We have already established that when a body moves with uniform acceleration along a straight line, let discuss about the motion under gravity,
a. v = u + ft, b. S = ut + ½ft²,
c. v² = u² + 2fS, and d. Sth = u + ½f(2t-1),
Where, u = initial velocity,
v = final velocity,
t = time during which velocity increases from u to v,
S = ut – ½ft², v² = u² – 2fs, and Sth = u – ½f(2t-1)
In case of vertical motion under gravity, ‘f’ will be replaced by ‘g’ and ‘S’ will be replaced by ‘h’.
Hence when a body falls under gravity, the following formulae will be applicable:
v = u + gt, …….(1)
h = ut + ½gt² ………(2)
v² = u² + 2gh ……….(3)
hth = u + ½g(2t-1) ………(4)
Where hth = height through which the body falls in the tth second.
When a body is thrown vertically upward, the following formulae are applicable:
v = u – gt, …….(5)
h = ut – ½gt² ………(6)
v² = u² – 2gh ……….(7)
hth = u – ½g(2t-1) ………(8)
TIME OF RISE OF A BODY PROJECTED VERTICALLY UPWARD
It has already been mentioned in a foregoing article of this chapter that when a body is thrown vertically upward, it is subjected to an acceleration, called acceleration due to gravity, which act vertically downward. This means that the body is subjected to a retardation ‘g’. As a result, the upward velocity of the projected body gradually decreases and an instant is arrived at when the velocity of the body becomes nil. From this instant, the body starts falling downward. Hence it is evident that the body projected vertically upward will go on rising until and unless its velocity becomes zero.
Late for a projected body,
T = required time of rise,
u = initial velocity with which the body is projected, and
g = acceleration due to gravity
Then, according to the formula v = u – gt, we get
0 = u – gT or, T = u/g
GREATEST HEIGHT ATTAINED BY A BODY PROJECTED VERTICALLY UP
Let, H = required greatest height attained by a body,
u = initial velocity with which the body is projected upward, and
g = acceleration due to gravity
At the greatest height the vertical upward velocity of the projected body becomes nil.
Hence, according to the formula
v² = u² – 2gh, we get
0 = u² – 2gH or, H = u²/2g
TIME OF FLIGHT FOR A BODY PROJECTED VERTICALLY UPWARD
Time of flight means time of rise plus time of fall of a body projected vertically upward.
When a body is projected vertically up motion under gravity, it goes on rising until its velocity is nil after which the body starts falling vertically downward and when it reaches the point of projection again, we can say that the displacement of the body is zero.
Let, T = required time of flight
u = initial velocity with which the body is projected upward, and
g = acceleration due to gravity
Then, according to the formula
h = ut – ½gt², we get
0 = u X T – ½gT² or, ½gT² = u X T.
Dividing both sides by T, we get
½gT = u or, T = 2u/g
TIME OF RISE = TIME OF FALL
Now, time of flight = time of rise + time of fall
2u/g = u/g + time of fall
or, Time of fall = 2u/g – u/g = u/g.
Thus, it is proved that for a body projected vertically upward, time of rise = time of fall.
TIME TO RISE TO A GIVEN HEIGHT
Let, u = initial velocity with which a body is projected vertically up,
h = height to which the body reaches (this is not the greatest height attained by the body),
t = time taken by the body to reach the height ‘h’, and
g = acceleration due to gravity
Then, h = ut – ½gt²
or, ½gt² – ut +h = 0 or, gt² – 2ut + 2h = 0
Thus, it is found that to reach the same height ‘h’, the projected body takes two times. This is because, when the body rises up, it reaches the given height ‘h’ in certain time, then again while in its downward motion (after reaching the maximum height) the time will be greater.
Evidently, during upward motion under gravity, time to reach the given height ‘h’ is smaller than the time taken by the body to fall to the same height during its downward motion.
Hence, during upward motion time to reach the given height ‘h’ = (u – √u² – 2gh)/g, and time to fall to the same height during downward motion = (u + √u² – 2gh)/g, both the times being measured from the instant of projection.
CONCLUSION
Motion under gravity refers to the behaviour of objects that move under the influence of the Earth‘s gravitational force. In the absence of air resistance, all objects experience the same constant acceleration of approximately 9.8 m/s². It is essential for understanding free fall, projectile motion, and planet and satellite orbits. Analysing motion under gravity provides insights into fundamental physics principles such as Newton’s laws of motion and the concept of gravitational potential energy.