Acceleration due to gravity
The Earth is a huge massive body. According to Newton law of
gravitation, the force of attraction between the two bodies is directly
proportional to the product of the masses and inversely proportional to the
square of the distance of separation. Here the two masses involved are the mass
of the body and the mass of the Earth. The distance between the bodies is
generally radius of the Earth, as the body is on the surface of the earth or
near to the surface of the earth. As the mass of the earth is comparatively
very high when compared with the body, the gravitational force of attraction
will be always towards the Centre of the Earth. This gravitational force
provides acceleration to all the bodies on the Earth and the corresponding
acceleration is called acceleration due to gravity. The numerical value of
acceleration due to gravity on the Earth is 9.8 m/s Squire. It may vary
slightly from place to place but in general it can be taken as a constant.
This acceleration on the surface of the earth is uniform and
constant. Hence when the body is coming towards the Earth, because of
acceleration due to gravity its velocity will be keep on increasing until it
strikes the ground. If you throw the body against the acceleration due to
gravity, into the sky, it’s velocity will be keep on decreasing and finally it
stops at a certain height called maximum height. Time taken to reach that
maximum height is called time of ascent and time taken to reach the ground from
a certain height is called time of descent.
We have four different equations of the motion to express the
translatory motion of a body. In all that equations where ever acceleration is
there, we can substitute acceleration due to gravity and we can rewrite the
corresponding equations of motion due to gravity. Here there are practically
two possibilities. The body may be coming towards the Earth or the body may be
going away from the Earth. The acceleration due to gravity on the bodies that
are coming towards the earth is generally treated as positive and vice versa.
We can write the equations of motion and the corresponding
values as shown below.
Using this equations we can find the time taken by the body
to reach a particular height against the gravity and it is called time of
ascent. If the body is falling from the same height to reach the ground to take
the same time and that is called time of dissent. The sum of the time of ascent
and time of dissent is called time of flight.
A body is said to be a freely falling body if it is falling
from a certain height with zero initial velocity. In that case, using the
equations of motion we can calculate the final velocity of the body after
covering a certain height h.
It can be also proved that if your body is thrown up with the
velocity from the ground, after reaching the maximum height it will come back
to the ground with the same velocity but in the reverse direction.
In general we have ignored the impact of the air resistance
while measuring the time of ascent and time of descent. If we consider the air
resistance, then the time of ascent is going to be little bit different from
time of dissent. The direction of the force due to the air resistance is always
in opposition to the relative motion.
So the effective acceleration in this case is going to be
more than acceleration due to gravity and hence the time of ascent is less than
that of the case when the air resistance is ignored. When we are ignoring air resistance
we are imagining that the environment is vacuum for the calculation purpose.
Though it is not practically vacuum the given equations will be approximately
valid in almost all the real-time situations.
By taking their resistance into consideration if we try to
calculate the time of dissent, being their resistances against the relative
motion, it is in the upward direction but the gravity is in the downward
direction. And hence the effective acceleration will be less and hence time of
dissent will be more and more than the case of vacuum.
It can be also be noticed that time of decent is more than
the time of recent when their resistances taken into consideration.
Problem and solution
Two balls are dropped
from different heights. One ball is drop two seconds after the other but both
the strikes the ground at the same time which is five seconds after the first
ball. Find the heights from where these two balls dropped?
As the first stone is taking five seconds to reach the ground
and the second stone is two seconds late, it shall be only taking three seconds
to reach the ground. The corresponding equations for the heights of the bodies
can be written basing on the equation of motion as shown below. Simply the
difference between the heights of the two equations we can calculate it as
shown below.
Problem and solution
A body balls freely
from a height of 125 m. After two seconds gravity ceases to act. Find the time
taken by it to reach the ground if acceleration due to gravity is considered as
10 metre per second Squire.
The body is a freely falling body means its initial velocity
equal to 0. For the first two seconds acceleration due to gravity is acting and
hence it falls due to the gravity and it covers a distance of 20 m. During this
process the body will acquire some velocity and it can be calibrated using the
equations of motion as shown below. It is found that its value equal to 20
meter per second. Being it is drop from 125 m and 20 m is covered in the first
two seconds, it has to further cover a distance of 105 m per reach the ground.
As acceleration due to gravity is no more acting the velocity
of the body will remain constant. Does the body will continue to move with the
same velocity of 20 m/s for the remaining time. We can calculate the time taken
to cover the 105 m is in the formula
that the displacement equal to velocity multiplied by time. By adding this time
of the two seconds with can get the total time of the journey.
Problem and solution
A parachutist drops freely from an aeroplane for 10 seconds
before the parachute opens out. He’s velocity when he reach the ground is 8
m/s. Once if the parachute is open he has a standard retardation of two meter
per second Squire. Find the height at which he gets out of the aeroplane.
Before the parachute opens he falls down due to the
gravitational effect and he’s not having a initial velocity in the vertically
downward direction. Hence we can find that is going to cover a distance of 490
m in the first 10 seconds. In this 10 seconds will also acquire some velocity
due to acceleration due to gravity and it can be found that the velocity is 98
m/s.
He covers the further distance with the standard retardation
of two meter per second Squire and we can calculate the distance that he covers
with a initial velocity of 98 m/s and a final velocity of 8 m/s and with an
acceleration of two meter per second Squire using the equation of motion. By
substituting the value is we can get the total height as 2875 m.
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