Motion Under Gravity

Motion under gravity of an objects is the influence placed on an objects in space by earth’s gravity. Last class we discussed about force definition and its calculations our today’s topic is motion under the influence of gravity.

motion under the influence of gravity

Motion Under Gravity

Motion under gravity : that is, motion under the influence of gravity, The motion of an object in space is influenced by the action of the earth’s gravity and air resistance.

If the air resistance, that is, force, is neglected, objects tend to have accelerated motion when falling from a given height or decelerated motion when projected upward into the air.
The expression above leads to three equations of motion under the influence of gravity which will be listed below.

Equation 1
V = u ± gt …….. (eqn 1)

Equation 2
V² = u² ± 2gh ……… (eqn 2)

Equation 3
h = ut ± ½gt² …….. (eqn 3)

Where h = height
g = acceleration due to free fall, a.k.a gravity and it is approximately to 10m/s² or 9.8m/s² (I remember talking deeply into how to derive S.I unit during our class session on derived and fundamental units.

Those are the three equations of motion under “motion under the influence of gravity” now let get the explanation: I mean what happens if an object is coming from upward? or what happens if an object is thrown upward?, do they have different acceleration due to gravity and do they have the same velocity or not?, well you will find out soon, keep reading.

What Happens When An Object Is Falling Down Under The Influence Of Gravity

So let praticalize this, play along please. Let first explain what happens when an object is thrown downward.

Let say you are standing on the veranda of a two storey building, and you wanted to give something, let say a ball to someone on the first storey, but you didn’t want to come down, so bit to the point you decide to drop the ball from the point where you are, that is, on the veranda of the two storey building to someone on the first floor.

As you drop or throw the ball from your point, if you are a good observant, you will notice that, the ball tends to fall more and more faster until it reaches that person on the first floor.

Ain’t you curious that, why is the ball moving faster, is someone pushing it? No, nobody is pushing it but something is thus pushing it, that thing is called gravity in full terms acceleration due to gravity.

You see, no matter which position you are now, either you are standing, sitting, walking, even lying down, etc it is the gravity that is keeping you in that position if not you will have been floating.

Meaning that, acceleration due gravity is the thing pushing you downward meaning that, acceleration due to gravity acts downward. Lol, some people do even make fun of it that “it is the gravity that reduces the height growth of human being.

If acceleration due to gravity tends to acts downward then what happens to the acceleration of our thrown ball.
Have it in mind that every moving object has its own acceleration. So meaning our thrown ball has two acceleration now, yes.

Since the acceleration of our thrown ball is moving in the same direction with acceleration due to gravity then the both accelerations are equal.
Hence, when an object is thrown or falling from upward:

Acceleration of the falling object = Acceleration due to gravity
Now let talk about its velocity,
As it has already been explain above, that an object falling from upward will tends to fall more faster and faster.
When you throw or drop an object from upward or let say whenever an object is falling from upward, the object has two velocity, one before it starts falling that is, the initial velocity and one after it has fell, which is the final velocity.

Notable Points : when an object is falling from upward, its initial velocity is zero and final velocity as maximum.

Hence,
Initial = u (m/s)
Final = v (m/s)

u = zero and,

v = maximum.

a = +g

Now let talk about what happens when an object is thrown upward.

What Happens When An Object Is Thrown Vertically Upward Under The Influence Of Gravity

Back to our practical above, also let assume that you are still standing on the veranda of that storey building and someone ones to throw you a ball.

It will be observed that, the velocity of the ball tends to decrease more slowly and slowly until it reaches you on the veranda.

Quickly think fast, what can you observe, when you throw the ball from upward and when the ball was thrown to you from downward.
Yes, the acceleration of the ball thrown to you from downward doesn’t move in the same direction with the acceleration due to gravity,

hence,
Acceleration of a moving object from downward = – Acceleration due to gravity

Because they didn’t move in the same direction.

Now let talk about the velocity of the thrown object from downward.

If you throw an object upward, you will apply more forces in order for the object to reach your desired destination, meaning that, the initial velocity of a vertically projected object will be maximum and its final velocity as zero.

Hence,
Initial velocity (u) = maximum
Final velocity (v) = Zero
a = -g.

That is the explanation on object in motion under the influence of gravity, now let take short note on the topic.

Acceleration Of Free Fall Of An Object Due To Gravity

Objects that fall freely have accelerated motion because of the action of the earth’s gravity influence on them.

As it has been explain above.
If a piece of stone is held, and let go from the top of a high-rise building, the object tends to fall from that height more and more faster until the object hits the ground.

Note : It is also observed that no matter, the difference in mass of objects, will all fall under gravity with the same acceleration and at the same time, only if air resistance is neglected. That is, a stone and a feather will both fall under the same acceleration and reach the ground at the same time.

LOL : I was explaining this topic to one funny candidate one day and he ask me that, how is that possible? And I said it is possible if the air resistance is neglected, that probably the physicist will have use a particular equipment to perform the experiment, and the candidate ask me that “what about inside the coffin (casket)?.

If air resistance is neglected acceleration due to gravity (g) can be determined by simple pendulum experiment or by free fall method.

Object Projection {Projectile}

Vertical Projection Of An Object Under The Influence Of The Earth Gravity

If a piece of stone is projected vertically upward, its projection velocity tends to decrease more slowly and slowly until it comes momentarily to a stop, and it then falls back faster and faster till it hits the plane of projection.

vertical projection of an object under the influence of gravity

The decrease in velocity as the object rises higher up is due to the effect of gravity. #because gravity is acting downward and the object is moving upward: negative direction#.

The maximum height of such vertical object under this influence can be determine, depending on the magnitude of the initial upward velocity used in projecting the object upward.

If h (maximum) = maximum height attained
u = initial upward velocity
g = acceleration due to gravity (free fall).

Then,
The time, t taken to attain the maximum height would be obtained from the first equation of motion.

v = u ± gt
And since a = −g

then
v = u − gt

Remember from the note above, at maximum height, the object usually comes to a momentary stop.

Hence,

v = 0
a = −g
then,
0 = u − gt
gt = u
 

t = u
     g
 

Note : the expression above is to calculate time to reach the maximum height not time of flight. Time to reach maximum height is represented by small letter t and time of flight is represented by capital letter T.

Determination Of Time Of Flight {T}

Time of flight is the time required for an object to be in the air or in flight.
Time of flight is twice the time required for an object to reach the maximum height.

Hence,

T = 2t
 

T = 2t
       g

Determination Of Maximum Height Attained

From the third equation of motion,

v² = u² ± 2gh
h = maximum height,
And at maximum height, v = 0,
0 = u² − 2gh
2gh = u²
 

h =
      2g
 

Horizontal Projection Of An Object Under The Influence Of Gravity

When an object is projected horizontally with an horizontal velocity, u, from a height, h, , it will have a half parabola path as shown in the diagram below and take the same time to reach the ground as though it is dropped vertically from the same height.

Determination Of Height, {h}, When An Object Is Projected

During the horizontal projection of an object, velocity in the vertical direction is zero.
By applying the second equation of motion:
h = ut + ½gt²
Remember that velocity along horizontal projection is zero, then:
h = ½gt² ,
then the time taken to reach the ground is:
h = ½gt²
2h = gt²

t = √2h
      g
 

Determination Of The Horizontal Distance {The Range} Covered With Horizontal Velocity {u}

Range is the horizontal distance covered by a projected object.
In horizontal distance direction, acceleration due to gravity is zero
Hence,
½gt² = 0
Inputting that into second equation of motion: h = ut + ½gt²
The range becomes,
h = ut
Where

t = √2h
     g
 

Then,

R = u × √2h
             g
 

Resultant Velocity Of A Projectile

When a projected object is in flight, its resultant velocity, v which is tangential to the direction of flight is made up of:

a] horizontal component velocity, Vx is equal to horizontal velocity of projection, U.

Vx = U

b] vertical component velocity, Vy is equal to vertical downward velocity, gt. {Since u = zero in the vertical direction and u = horizontal velocity of projection}.

resultant velocity of a projectile

Horizontal Projection Of Ball Indicating Velocity After Time, t

The resultant velocity, v after time, t, v = √Vx² + Vy²

And since Vx =u and Vy = gt
v = √u² + (gt)²

If V is inclined at an angle θ to Vx
then,

tan θ = Vy
             Vx
 

= gt
   u
 

Motion Down An Inclined Plane

If the friction between an object and the inclined plane in which it is in contact with, is neglected, the object would tend to slip down the inclined surface faster because of the component effect of acceleration due to gravity.

motion down an inclined plane

Resolution Of An Inclined Plane

Component of g along the plane, g, gx = sin θ and that perpendicular to the plane, gy = g cos θ.
if s = length of the inclined plane,
t = time taken to slip down the plane,
θ = angle of inclination of the plane
v = final velocity attained on reaching the base of the plane,
u = 0 {at rest},

Hence,
Final velocity, v =gxt, when u = 0

I.e v = gt sin θ

Distance covered, s = ½gxt²

I.e s = ½gxt² sin θ

if v² = u² + 2gs is applied,
then,
Since u = 0

s =
     2gx
s =
   2g sin θ

Determination Of Vertical Height Of Falling Object On An Inclined Plane

determination of vertical height of free fall

sin θ = h
             s
Hence,
h = s Sin θ

Remember that,

s =
      2g sin θ

Then substitute,

h = × Sin θ
2g sin θ
I.e

h =
      2g
 

Now let talk about Inclined Projection.

Inclined Projection

inclined projection

When an object is projected into space at an angle θ° to the horizontal, five things happen,

a] A fully parabolic path is drawn as shown in the diagram above.

Example of parabolic motion is: a thrown javelin or discus into space or an arrow shot into the air, or when a missile or a rocket is launched into the space.

b] It velocity of projection decreases slowly and slowly as it gains height.

c] The velocity becomes zero, at maximum height reached.

d] Due to gravity, the projected object returns to the same projection plane with a distance R from the point of projection.

e] The vertical component of velocity ux = u sin θ and,
The horizontal component velocity uy = u cos θ

Important note : an object that moves through space on projection is called a Projectile and the path taken is called Trajectory.

Determination Of The Time Taken To Attain The Maximum Height In An Inclined Projection

a] Vertical component velocity = uy
Final velocity at maximum height = v = 0
Acceleration of free fall = −g
From v = ux − gt
0 = uy − gt
Then,

t = uy
    g
Hence,

t = u sin θ
   g

Determination Of Time Of Flight In An Inclined Projection

Time of flight, T is the time required for a projected object to be in flight and it is twice the time required to attain the maximum height, H

I.e T = 2t
And

t = u sin θ
  g
Hence,

T = 2u sin θ
   g

Determination Of Maximum Height Attained In Inclined Projection

From the third equation of motion,
v² = u² − 2gh
And v = zero, at maximum height
Then,
0 = u²y − 2gH
2gH = u²y
Recall that, uy = u sin θ

Hence,
2gH = (u sin θ)²

H = u² sin θ²
    2g

Determination Of Range In Inclined Plane

The Range {R} is the horizontal distance from the point of projection to the point where the projected object hits the horizontal plane.
The velocity that enables the projectile to cover this range is the horizontal component velocity, ux = u cos θ

Since g = 0, along this direction and time required to cover the range is equal to time of flight, T,
Then,
R = ux,

T = u cos θ × 2 u sin θ
                    g
Hence,

R = 2u²sin θ cos θ
           g
But sin 2θ = 2 sin θ cos θ

Hence,

R = u² sin 2 θ
   g
 

Important Note : The maximum Range a projectile can attain is when the angle of projection is 45°

Then,

R = u² sin 2 θ
   g
R = u² sin 2 × 45
                 g
Since sin 90° = 1
Hence,

R =
      g

Wow.. wuuuhh, that is a long trip, we have almost come to the end, let discuss about it applications and next class, we will be doing some calculations on motion under the influence of gravity.

Application Or Uses Of Projectile

1] Projectile is applied during the shooting of arrows, guns and rockets.

2] Launching of mechanically propelled missiles, such as inter-continental ballistic missiles.

3] Projectile is also applicable in sport and gaming, e.g throwing of discus and javelin, throwing of basketball, kicking of football and volleyball, etc.

4] It is also applied in taking-off and landing of aircrafts.

And now we come to the end of motion under the influence of gravity and projectile, till next class where we’ll be solving calculations on the topic, till then GOD bless me. Don’t forget to write a comment and hit that social sharing button below, thank you.


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