Hello friends, I hope you are all well and doing your best in your fields. In this post, we can explore the fundamental concept in physics: projectile motion. Projectile motion is the motion of the moving particle or the moving body that can be projected or motion near the earth's surface. Still, the particle can be moved according to the curve path or under the force of gravity and the gravity line. In history first, galileo represented particle motion in the form of projectile motion which can occur in the form of the parabola( the u-shaped curved or mirror-symmetrical in which the particle can be moved) or the motion of the particle which may occur in a straight path in the like if the ball throw downward from upward their motion path is straight.
The detailed or fundamental concept of projectile motion is essential to understand in different fields like mechanics, astronomy, or military sciences because it can help to understand the motion of rockets that can be used in wars. If the rocket can be launched from the earth to the next point it can do the projectile motion because they can be moved on the parabola. Now in this article, we can discuss and explore the projectile motion, its introduction, definition, mathematical representation, applications, numerous problems, and their significance.
Projectile motion can be defined as:
“The two-dimensional motion of the moving particle or the object with their inertia, and under the constant acceleration or the gravitational force is termed as projectile motion.
Some examples of trajectory motion are given there:
When the footballer player kicks the ball from one point then the ball follows the parabola and reaches the other this is the trajectory motion.
The bullet can be fired from the gun.
The ball can be thrown from an upward to a downward direction
The rocket or the missile can be launched and moved toward space under constant acceleration or the force of gravity.
The trajectory is defined as:
“The path which can be followed by the projectile motion particle or object is termed the trajectory. The path that can followed by the projectile particle are parabola so their trajectory is the parabola.”
The parabola is the curve in which the projectile motion occurs and their curve is mirror-symmetrical or may be like the u- shaped. In parabola two dimensional motion can occur and it can occur in the dimension of x and y.
The equation or formula of the parabola is written below:
In the dimension of the x-axis:
y = a ( x -h)2 + k
There,
a represents the constant acceleration, and h represents the height but in this equation, both h and k are the vertexes of the parabola.
In the dimension of the y-axis:
y = a ( y -k)2 + h
There,
a represents the constant acceleration, and h represents the height but in this equation, both h and k are the vertexes of the parabola.
Ballistic is defined as:
“The study of the projectile motion is termed as the ballistic and the study of the projectile motion trajectory are termed as the ballistic trajectory.”
The fundamental explanation of the projectile motion with their basic principles ( horizontal motion, vertical motion ) is given there:
The motion of an object in a horizontal direction:
When the body or the ball can be thrown from upward with the angle or the initial velocity then it can be moved forward because of the moving body inertia and falls downward because of the constant gravitational force acting on it. So according to this, in the horizontal direction of motion, no forces acted on it (only gravitational force act on it) so that is why the acceleration in the horizontal direction is equal to zero as,
ax = 0
The motion of an object in a vertical direction:
When the body or the ball can be thrown from upward with the angle or the initial velocity then it can be moved forward because of the moving body inertia and falls downward because of the constant gravitational force acting on it. According to this, in the vertical direction of motion, forces acted on it so that is why the acceleration in the horizontal direction is equal to g, and g = 9.8ms2 .
The path of the trajectory can be determined through the given equation, their derivations are written below:
As we know the second equation of motion,
S = vit + 12at2
There,
vi represent the initial velocity, a indicates the acceleration and t represents the time.
In the x dimension, we can write this formula as:
x = vixt + 12at2
As we know, in the x dimension the acceleration is equal to zero so,
ax = 0
x = vixt + 12(0)t2
So,
x = vixt + 0
x = vixt
In the y dimension, we can write this formula as:
y = viyt + 12at2
As we know, in the y dimension the acceleration is equal to g so,
ax = -g
y = viyt + 12(-g)t2
So,
y = viyt - 12gt2
In some special cases when the projection of the moving body is projected horizontally from some certain height then,
y = viyt - 12gt2
Then,
viy = 0
y = (0)t - 12gt2
y = 12gt2
Consider the projected body that has the initial velocity vi and at the horizontal direction the angle θ can be formed between them so the initial velocity for horizontal or vertical components is equal to cos or sin, their equation is written below:
Initial velocity for the horizontal component = vix= vi cosθ
Initial velocity for the vertical component = viy = vi cosθ
Their detailed derivation is given there:
On the horizontal dimension moving object, no force acts on it only gravitational force acts on it so that's why the acceleration is equal to zero and written as:
ax = 0
As we know the first equation of motion
vf= vi + at
So the velocity for the horizontal component in the x dimension is written as:
vfx= vix + axt
ax = 0
So,
vfx= vix + (0)t
vfx= vix + (0)
vfx= vix or it also equal to,
vfx= vix = vi cosθ
On the vertical dimension of moving objects, the forces acting on it or the acceleration are equal to g,
ay = -g
As we know the first equation of motion
vf= vi + at
So the velocity for the vertical component in the y dimension is written as:
vfy= viy + ayt
ay = -g
So,
vfx= viy + (-g)t
Or,
viy = vi cosθ
So,
vfx= vi cosθ - gt
The magnitude can be determined for the components that can be moved in two dimensions. The formulas which are used for determination are given there:
v= vfx2 + vfy2
There,
v represented the velocity of the components, vfx represented the final velocity for the x components, and vfy represented the final velocity for the y components.
In the two-dimensional components, the resultant velocity can form the angle θ between their horizontal components the formula for determining their direction is given there:
tan Φ = vfyvfx
Or,
Φ = tan-1 vfyvfx
The displacement can covered by the projectile object in the time t so the displacement in the horizontal or the vertical component can be written as:
x = vixt cos θ
y = viyt sinθ - 12gt2
So, to find the magnitude of the two dimension body displacement we can use the given formula:
Δ r = x2 + y2
Now let the both equations as:
x = vixt cos θ, y = viyt sinθ - 12gt2
Then, eliminate the time from the above equation and write them as,
y = tan θ. x - g2v2 cos2θ . x2
So, we know that
R = g2v2 cos2θ
R indicates the range of the projectile motion
So,
y = tan θ. X - x2R
The g, angle is x2so it can also be written as,
y = ax + bx2
This equation or the formula can slo be used for parabola but the angle can be formed and this equation can be written as,
v = x2gx sin2θ - 2y cos2θ
Displacement of the components can also be shown in the polar coordinate system or in the cartesian coordinate system. For the determination of the displacement in the polar coordinate system, we can use the given formula which is written below:
r ф = 2 v2 cos2θg (tan θ secф - tan ф secф )
According to the above equation or derivation, we know that,
y = r sinθ
or , x = r cosθ
There are some basic properties of the projectile motion or the trajectory which are given there:
The maximum height of the projectile object is when the projectile object can reach the highest point or the projectile object covered the maximum distance to reach the peak is termed as the maximum height of the projectile object.
To determine the maximum height of the projectile motion we can use them,
The initial velocity for the projection of the object = viy= initial velocity in the vertical component = viy = vi sinθ
So we can also know that the acceleration in the vertical velocity the acceleration is equal to g
ay = -g
Or the final velocity when the projectile object can be reached at the maximum height,
vfy = 0
So,
= v sinθ - gth
So the time that can used to reach the maximum height,
th= 2v sinθg
There, th indicates the time of the projectile motion to reach the maximum height.
As we know,
2aS = vf2 - vi2
Or this equation can be written as,
2ayy = vfy2- vfx2
This equation is used for the vertical component
Now put the values in this equation and write them as
2(-g) H = (0) - ( visinθ)2
Then,
-2gH = vi2 sin θ2
Then, the height of the projectile motion can be determined by,
H = vi2 sin θ22g
There, H indicates the height of the projectile motion of the moving objects.
When the maximum height is reached then the sin θ = 90°
Hmax = vi2 ( 0)2g
Hmax = vi2 2g
So the maximum height when the angle formed between the vertical and the horizontal components we can use the given formula:
H = (x tanθ)24 ( xtan θ -y)
Now to find the angle of the elevation at the maximum height we can determine this by using the given formula which is written below:
Φ = arctan tan θ2
the maximum distance that can be covered by the projectile body in the horizontal direction is termed the range of the projectile.
To determine the range of the projectile in the horizontal direction we can use the given formula that can be derived from different equations so the derivations are given there:
As we know,
x = vix t + 12 axt2
So,
vix = vi cosθ
t = 2v sinθg
ax = 0
x = R
So, according to this,x = vix t + 12 axt2, this equation can be written as,
R = vi cosθ 2vi sinθg + 12 (0)t2
R = vi cosθ 2vi sinθg + 0
R = vi2 ( 2sinθ cosθ)g
We also know that 2sinθ cosθ = sin 2θ
R = vi2 sin 2θg
The relationship between the maximum height and the horizontal range can be proved through the given derivation and formula which are given there:
As we know,
H = vi2 sin θ22g
We can also know that,
d = vi2 sin2 θ2g
Then we can compare both of these equations to prove the relationship between them,
hd = vi2 sin2θ2g gvi2 sin2 θ
hd = sin2θ4 sinθ cosθ
So,
H = d tan θ4
Then, the height of the projectile can equal the range of the projectile of the body
H = R
The time of flight of the projectile body can be defined as the time that can be used to cover the distance from their launching to reach the end where the projectile body can be taken off. Simply the time that can be used for the moving projectile body to hit the ground is termed as the time of flight of the projectile body.
When the projectile body starts initial velocity can go up but again come back to the ground with the same velocity so it cant cover the vertical distance we know that the vertical distance is equal to zero and written as:
y = 0
So we know that,
The initial velocity which is used by the projectile body = viy = vi sin θ
The acceleration in the vertical velocity which is due to the force of gravity ay = -g
Then we can determine the time of the flight by using the equation which is given there:
S = vit + 12 at2
Then put these values or rewrite the equation as;
y = viyt + 12 ayt2
Then,
0 = ( vi sinθ) t - 12 gt2
12 gt2 = ( vi sinθ) t
t = 2vi sinθ g
According to the given equation, we can eliminate the air resistance but if the time of the projectile body vertical direction with the height at 0 then it can be written as:
t = dv cos θ
There, d represented the displacement. So it can be written as:
t = v sinθ + ( v sin θ2) + 2gyg
Now solve this equation as
t = v sinθ + ( v sin θ2) + 0g
t = v sinθ + ( v sin θ2) g
Then eliminate the by the 2 power and write them as
t = v sinθ + v sin θg
t = 2v sinθ g
If the θ = 45°
Then put this value in the equation
t = 2v sin(45)g
t = 2v sin22g
t = 2vg
The projectile body can reach the maximum range when the sin 2θ reaches the maximum value because sin 2θ = 1 there, to find the maximum range we can use the given formula and determine them. Their formula with derivation is written below:
As we know,
Sin 2θ = 1
2θ = sin-1 (1)
Or, sin-1 (1) = 90°
2θ = 90°
But, θ = 45°
We can also that,
R = vi2 sin 2θg
Then put the value of θ
R = vi2 sin 2(45)g
R = vi2 sin (90)g
sin 90° = 1
R = vi2 (1)g
R = vi2 g
The maximum range of the projectile motion can be written as the:
R = Rmax = sin 2θ
Ballistic is defined as:
The study of the motion of the projectile body is termed as the ballistic.
Detailed exploration of the ballistic is given below:
Ballistic flight can be defined as:
The projection of the body starts when an external force is applied or can ut the initial push then the object can be moved freely without any restriction or the object move with inertia or also due to the force of gravity that can act on the projectile body this types of flight are termed as the ballistic flight.
Ballistic missiles are the type of ballistic flight in which the missile can do projection with un-powered or also with un-guided. Ballistic missiles are used in the wars by the military or also in astronomy.
The path or the curve that can followed by the ballistic missile or the ballistic flight is termed as the ballistic trajectory.
A ballistic missile can follow the ballistic trajectory but the missile or the flight can be moved due to the two independent motions through which the body can be moved freely and reach its destination. The two main or independent positions are given there:
The force of gravity and the inertia of the body help the object to move or follow the parabolic path which can do the projectile motion or the ballistic flight. Both of these forces are essential for the free motion of the projected body and reached to their destination.
The projectile body can fly or in starting follow the strength path in the direction of launching and then follow the parabolic path or do the projectile trajectory.
Interia is the force that can help the body to move straight with the force of gravity. But with the force of inertia, the projectile body can move straight or fall to the point where its destination is fixed or reach the point where it can be thrown down. However, due to the effect of inertia, the constant speed or the velocity is always equal to the initial speed or the velocity in space.
When the body can be moved it can do a straight motion due to the effect of inertia but the trajectory path or the parabola path can be followed by the due to the force of gravity. Because the force of gravity turned the body or the object to move in teh curved space and helped to attract into the ground and reach its destination.
For the short-range motion or if the motion reaches the earth then the projectile body always follows the parabolic path due to the effect of inertia and the force of gravity.
The long-range motion of the projectile body or the projectile body that can be moved in the spherical earth is termed elliptical.
This trajectory path is mostly followed by missiles which are used in wars or also used when rockets or missiles are launched.
Some major uses of ballistic missiles are given there:
Short ranges: ballistic missiles or ballistic trajectors are mostly used for short ranges they are not used mostly for long ranges.
Long ranges: for the long ranges ballistic missiles or ballistic trajectories are used but these can used by controlling them from remote and also launching these missiles by providing complete guidance to them.
Air friction: when the trajectories are moved with a high velocity then the air resistance can't be neglected it can calculated with the air friction. Because mostly the air friction in the atmosphere or space is greater than the force of gravity that’s why it can't be neglectable.
Aerodynamic forces: when teh force of gravity becomes less according to the air resistance and it affects both horizontal or vertical component motion so then we can't neglect the aerodynamic forces which are mostly air resistance.
Aerodynamic forces can affect the projection directly because the air resistance can create many different problems in the flight so for the projectile motion, the moving projectile body needs a high level of the projection angle to move efficiently.
The factors that affect the motion of the projectile bodies are given there:
Air resistance
Initial velocity
Height of launch
Angle of projection
Now in the calculation of the projection of the projectile bodies, air resistance can't be neglected because air resistance and other aerodynamic forces can affect the projectile bodies' projection, height, and ranges.
The initial velocity can directly affect the projectile motion because if the initial velocity is high then the projection and the height of the flight are also high and reach their destination with the high velocity and speed.
When the projectile body moves or is launched at a high height then its range and the time that can be taken by it to be thrown are increased because its height or range with the angle of projection are increased.
The angle of projection directly affects the range and the height of the projectile body because if the angle is increased then they have a high projection, the optimal angle of projection is 45 if we neglect the air resistance then at this angle, the body can be reached at its maximum height.
Some major applications of the projectile motion are given below:
Space exploration: understanding and analyzing the projectile motion can help in space exploration to study the stars and galaxies.
Engineering: understanding and analyzing the projectile motion can help in engineering to manufacture the rockets and missiles which are used in teh wars or used in space exploration.
Sports: projectile motion also helps in sports like when we use a gun then the projectile motion concept is essential to understanding teh process of fire.
Military: in the military projectile motion is fundamental because the rockets and the missiles being used move according to the trajectory path which is understood after clearing the concept of projectile motion.
Some applications of the projectile motion in the advanced topics are given there:
The motion of the projectile body in non-uniform gravitational fields.
Air resistance
Drag force
Spin and Magnus effect
To study the projectile motion or the projectile trajectory through experiment the engineers can use different types of machines or instruments like motion sensors, tracking software, or different types of high-speed magnification cameras and lenses to see or analyze the trajectory path of the projectile body and through analyze they can improve the theoretical model which re based on the projectile motion. Experimental studies of the projectile motion help to precise or accurate the different models and also help to understand their applications in different fields.
In different fields of physics, mostly in mechanics or astronomy projectile motion is used to understand the motion of the projected objects and also help to understand the motion or the trajectory path because, in projectile motion, motion is affected by the force of gravity and inertia also. In the projectile motion, we can analyze the path, range, and maximum height of the projected objects precisely and accurately. After understanding these basic properties and the principle of the projectile motion we can use this in different fields like in engineering or mainly in the military. Now modern or advanced topics like air resistance or the different forces effects can be analyzed easily through understanding the projectile motion. After reading these articles the reader can understand the projection of the projectile motion efficiently.
Hi friends, I hope you are all well. In this post, we can discuss the fundamental concept of collision crucially. Generally, collision is the interaction between two moving bodies because when two bodies interact then they can change their direction during the motion. In physics, we can deal with and understand the motion of the moving bodies so collision is a force that can exert the moving bodies when two or more bodies come in contact for a short period. In moving bodies when two bodies collide they can exert a high force and collide with each other with great force but in their collision, the kinetic energy always remains conserved.
When the collision occurs between the two objects, it can change their velocity because they can change their direction and move quickly. The change in the velocities after collision has a high difference and it can also be termed as the closing speed. kinetic energy is always conserved so that's why they also conserved the momentum. In atoms, the inside particles or all subatomic particles can also collide so to understand their collision it is compulsory to understand the types of collision and their significance. In the field of mechanics, kinematics the concept of collision is fundamental to understanding it. Now we can start our detailed discussion about the collision, its types, elastic collision, inelastic collision, special cases, examples, and their different natural phenomena.
Collision is defined as:
“When the two particles collide with each other by exerting a high force, maybe their collision occurred accidentally but the forceful interaction between the two moving bodies or particles is termed as the collision.”
The collision can't be perfect because only in the ideal gases perfect collisions may be occurred but mostly perfect collisions aren't possible. The collision can mostly occur in gases or liquids or atoms because it can only occur when the free particles are present and do continuous or random motion their motion is not steady. Because in steady motion between two particles collision cant be occurred.
The general formula that can be used for the collision between two bodies is written as:
m1v1 + m2v2 = m1v1' + m2v2'
There,
When the ball bounces on a hard marble floor then it can also bounce back because it can collide with a hard surface momentum and the kinetic energy remains conserved but if the hard ball can bounce on a soft surface or the sandy surface then it can't bounce back and this collision of the ball with the sandy surface are inelastic collision because it can't bounce back and the elastic collisions are those in which the ball bounces back again.
When cricketers or football play a game on the field they can collide with each other with a great colliding force.
The car which can be moved on the road with high speed and velocity and suddenly collide with the other car then both collide with the high velocity or speed and exert the high colliding force.
Collison has common two types but they have three major types which can be written below with the detailed description and examples:
Perfect inelastic collision
Inelastic collision
Elastic collision
Elastic collision is defined as:
“ when kinetic energy is conserved during the collision between the two moving particles or objects termed as elastic collision”
In this type of collision, always momentum and energy remain conserved. Elastic collisions are ideal because in this collision the kinetic energy of the colliding objects remains the same before the collision and after the collision. In surroundings rarely elastic collisions can be seen because they are ideal so that's why they can generally seen in between atoms or in between the subatomic particles or molecules.
In elastic collisions, the energy is conserved when no heat or sound energy can be produced. But the perfect elastic collision is not possible. when the two bodies collide with each other with great force firstly energy is converted from kinetic to potential then the particles again start moving then they again convert the potential energy into kinetic energy by creating the repulsive forces and by making the angle between their collision. Through this, the moving particles can conserve their energy. The elastic collision of the atoms can firstly shown by the rutherford through his atomic model. In the concept of elastic collision, the bodies that can collide with each other have the same mass so they can conserve both momentum and kinetic energy without releasing any energy in the form of heat, sound, or other. Elastic collisions only occur during the random or variable motion of the atoms or bodies like when the atoms of gases collide with each other then it can be shown the ideal elastic collision which is not possible.
When the hard ball hits the hard surface then it can bounce back with the same velocity because it can be shown the elastic collision in which the momentum and the kinetic energy are remained the same before and after the collision.
In elastic collision with the kinetic energy, the momentum can also be conserved so that is why it is important to understand the law of conservation of momentum. The simple statement in which the law of conservation can be defined is given there:
“The body that can be moved with linear motion, then the total momentum during their linear motion of the isolated system ( the system in which no external force can be exerted) can always remain constant.”
Mathematical representations of the law of conservation of momentum are written below:
m1v1 + m2v2 = m1v1' + m2v2'
There,
m1 and v1 represented the mass and the velocity of the first moving object and m2 or v2 the mass and velocity of the other object that can collide with the first object.
m1 and v1' represented the mass and velocity of the first object after the collision and m2 and v2' indicate the velocity of the second object after the collision.
To understand the elastic collision in one dimension let's suppose the moving bodies or the hard balls which are non-rotatable and have equal masses. Their masses can be represented through m1 or m2 and their velocities before collision are represented through v1 and v2, but when these two balls collide with each other their mass remains the same as the m1 or m2 but their velocity is changed, and represented as v1' or v2'.
According to the above explanation, we know that m indicates the masses of the bodies and v indicates the velocities of the objects now it can be mathematically represented through the law of conservation of momentum and it can be written as:
As we know the law of conservation of momentum,
m1v1 + m2v2 = m1v1' + m2v2'
Then, when we arrange them and write them as,
m1v1 - m1v1' = m2v2' - m2v2
Or, when we take the m1 or m2 common then it can be written as:
m1( v1- v1' ) = m2 (v2'- v2) ………. (i) equation
We know that the elastic collision is the perfect elastic so in this collision, the kinetic energy is conserved totally and it can be written as:
12m1v12 + 12m2v22 = 12v1v1'2 + 12m2v2'2
Now arrange them according to their masses and write as
12m1v12 - 12v1v1'2 = 12m2v2'2 - 12m2v22
Now take the common m1, m2,
12m1 (v12 - v1'2 ) = 12m2 (v2'2 - v22)
Now cut the same value 12 on both sides and write as
m1 (v12 - v1'2 ) =m2 (v2'2 - v22) ……. (ii) equation
Now divide the equation (ii) from the equation (i) and write as
m1( v1- v1' ) = m2 (v2'- v2) ………. (i) equation
m1 (v12 - v1'2 ) =m2 (v2'2 - v22) ……. (ii) equation
Then,
m1 (v12 - v1'2 ) m1( v1- v1' ) = m2 (v2'2 - v22)m2 (v2'- v2)
As we know,
( v12 - v1'2 ) = ( v1 - v1' ) ( v1+ v1')
(v22- v2'2) = ( v2 - v2' ) ( v2+ v2')
Then we can put these equations in the above equations cut the same masses and write them as,
( v12 - v1'2 ) ( v1- v1' ) = (v2'2 - v22) (v2'- v2)
( v1 - v1' ) ( v1+ v1') ( v1- v1' ) = ( v2 - v2' ) ( v2+ v2') (v2'- v2)
Then,
v1 + v1' = v2 + v2'
Then arrange their velocities before and after the collision and write them as,
v1 - v2 = (v2' - v1')
Arrange them and write them as
v1 - v2= - (v1' - v2')
Now the given equation which is used for the elastic collision in one dimension shows that ( v1 - v2) shows the magnitude of the relative velocity of the 1st ball as compared to the second ball before the collision.
And v1' - v2' shows the magnitude of the relative velocity of the 1st ball as compared to the second ball after the collision.
And this represented that,
Speed of the ball approach = speed of the ball's separation.
For the final velocity of the moving particle according to the newton we can use the given formula:
v = ( 1+ e) vcom - ev
Or, v = vcom = m1 v1 + m2 v2m1 + m2
There,
vcom represented the two particles' center of mass related to velocity.
e represented the coefficient of the velocity restitution.
v is the initial and the final velocity which can be different before the collision or after the collision.
The formula of the special relativity that can be used in the relativistic velocity in one dimension, using the relativity formula is written below:
ρ = mv1 - v2c2
There,
ρ represented the momentum, m indicates the mass of the moving particle v represents the velocity and c indicates the speed of light. but according to this formula, the total momentum of the moving particles is equal to zero. And their description is written.
ρ1= -ρ2
So that is why, ρ12 = ρ22
And the E is equal to,
E = m12c4 + p12c2 + m22c4 + p22c2
Then,
v1 = -v1
After the collision, the velocity can be calculated by using the equations of the moving objects or the particles. the details and formulas that can be used for the determination are given there:
We can determine the velocity of the mass after collision by using the formula derivation and formula are given there:
As we know,
v1 - v2 = v2' - v1'
Then,
v2' = v1 - v2 + v1' ….. (i) equation
We also know that
m1( v1- v1' ) = m2 (v2'- v2)..... (ii) equation
Now put the equation (i) into the equation (ii)
m1( v1- v1' ) = m2 (v1 - v2 + v1'- v2)
m1v1 - m1v1' = m2v1 - m2 v2 + m2v1' - m2v2
Then arrange them,
m1v1' + m2v1' = m1v1 - m2v1 + m2v2 + m2 v2
Then,
v1' ( m1 + m2) = v1( m1- m2) + 2 m2 v2
Or,
v1' = v1 ( m1- m2)( m1 + m2) + 2m2(m1 + m2)v2 ……. (iii) equation
The above formula can be used to find the velocity of the mass after the collision.
To find the velocity of the second mass after the collision we can use some equations their derivation is written below.
Now use the equation (i) and equation (iii)
v2' = v1 - v2 + v1' ….. (i) equation
v1' = v1 ( m1- m2)( m1 + m2) + 2m2(m1 + m2)v2 ……. (iii) equation
Now put the equation (iii) into the equation (i)
v2' = v1 - v2 + v1 ( m1- m2)( m1 + m2) + 2m2(m1 + m2)v2
v2' = v1 1 +( m1- m2)( m1 + m2) + 1-2m2(m1 + m2) v2
v2' = v1 ( m1- m2) + (m1+ m2)( m1 + m2) +v2 2m2 - (m1 + m2)(m1 + m2)
Then,
v2' = v12m1( m1 + m2) + v2 m1- m2 m1 + m2 …….. (iv) equation
There are some special cases in which the masses become equal or some are not equal but they have some target mass and their collision depends on them. Some cases are discussed below:
In the first case, the mass of both bodies m1 m2 is equal so that is why the is a mass exchange of both moving velocities after the collision.
To prove the above case we can use the equation (iii) and the equation (iv) which are given there.
Firstly we can use the equation (iii)
v1' = v1 ( m1- m2)( m1 + m2) + 2m2(m1 + m2)v2
According to this case, we know that m1 = m2 so,
v1' = v1 ( m- m)( m + m) + v2 2m(m + m)
v1' = v102m + v2 2m2m
v1' = 0 + v2
v1'= v2
According to this equation the velocity of the second mass exchange with the velocity of the first mass after collision.
Then use the equation (iv)
v2' = v12m1( m1 + m2) + v2 m1- m2 m1 + m2
In this case, we can also equal the m1 = m2 so,
v2' = v12m( m + m) + v2 m- m m + m
Then,
v2' = v12m2m + v2 0 2m
v2' = v1 +0
v2' = v1
According to this equation the velocity of the first mass exchange with the velocity of the second mass after collision.
In the second special case, the mass of both bodies is equal but the velocity of the second mass is also equal to zero.
To prove the above case we can use the equation (iii) and the equation (iv) which are given there.
Firstly we can use the equation (iii)
v1' = v1 ( m1- m2)( m1 + m2) + 2m2(m1 + m2)v2
According to this case, we know that m1 = m2, or v2= 0
v1' = v1 ( m- m)( m + m) + 0 2m(m + m)
v1' = 0 + 0
v1' = 0
According to this equation, the velocity of the second mass can be used by the first mass.
Then use the equation (iv)
v2' = v12m1( m1 + m2) + v2 m1- m2 m1 + m2
In this case, we can also equal the m1 = m2 or v2 = 0
v2' = v12m( m + m) + v2 m- m m + m
Then,
v2' = v12m2m + 0
v2' = v1
According to this equation, the velocity of the second mass after collision is equal to the velocity of the first mass before collision.
An elastic collision can occur in two dimensions. The motion or the elastic collision can be determined or understood through the law of conservation of momentum or the conservation of kinetic energy with the angular momentum. In the two-dimension collision, the first collision can occur in the ball line and the other occurs when the proper two moving bodies can collide hard. During this type of elastic collision, an angle can be created between them.
Derivation in which the two moving objects can collide with each other during the motion in two dimensions on the x-axis and the y-axis are given there:
To determine the elastic collision in the x-axis we can use the given formula,
v1x' = v1 cos (θ1 - φ) ( m1 - m2 ) + 2m2v2 cos (θ2 - φ)m1 + m2 cos θ + v1 sin (θ1 - ဖ ) cos φ + π2
To determine the elastic collision in the y-axis we can use the given formula,
v1x' = v1 cos (θ1 - φ) ( m1 - m2 ) + 2m2v2 cos (θ2 - φ)m1 + m2 sin θ + v1 sin (θ1 - ဖ ) sin φ + π2
These formulas can be used to determine the x and y-axis dimension motion of the bodies but if these motions can occur without the angles these formulas can be written as:
v1' = v1 - 2m2m1+ m2 (v1 - v2 , x1 - x2x1 - x22) (x1- x2)
Or,
v2' = v2 - 2m2m1+ m2 (v2 - v1 , x2 - x1x2 - x12) (x2- x1)
For determination of the angle between in two-dimensional collision, we can use the given formula which can be written below:
tan θ1 = m2 sin θm1 + m2 cos θ
or,
θ2 = π - 02
To determine the magnitude of the two moving bodies in two dimensions we can use formulas which are written below:
v1' = m12 + m22 + 2 m1m2 cos θm1 + m2
Or,
v2' = v1 2m1m1 + m2 sin θ2
Inelastic collision is defined as:
“The kinetic energy that is not conserved during the collision is termed as the inelastic collision.”
In this type of collision the kinetic energy can be changed into other forms of energy due to the friction that can be produced when the two moving bodies collide hard and their kinetic energy can be changed into heat energy, sound energy, and potential energy.
Inelastic collisions can be mathematically represented through the given equation.
m1 v1i + m2v2i = m1v1f' + m2 v2f'
Now, we know that in this type of collision kinetic energy cant be conversed so that's why it can be changed into different types of energy so it can be represented through the given equation which is written below:
12 m1 v1i2 + 12 m2 v2i2 12 m1 v1f2 + 12 m2 v2f2
There are two main types of inelastic collision which are given there;
Perfectly inelastic collision
Partially inelastic collision
In a perfectly inelastic collision, the two moving bodies that collide with each other are stuck together when they come closer for collision or they can't collide like the elastic collision. In this type of collision, the kinetic energy that is not to be conserved changes into other forms of energy totally as sound energy, heat energy, potential energy, and others.
A perfectly inelastic collision can be represented through the given equation:
m1 v1i + m2v2i = (m1+ m2 ) vf'
Through this equation, it can be proved that the final velocity after the collision is the same for both masses because both moving bodies can be stuck together.
In this type of inelastic collision, the moving bodies or masses can't stuck together but in this collision, most of the kinetic energy can not be conserved and change into different forms of energy but some kinetic energy may be conserved. In the real world or our surroundings, partially inelastic collisions occur because this type of collision is in the real world.
The some major examples of the inelastic collision are given there:
The car that can move on the road can collide with the other car then the kinetic energy that is produced during motion can be conserved somehow but mostly can be changed into another form of energy like heat energy, sound energy, and potential energy.
When the ball can collide with the soft floor then there kinetic energy can't be conserved so that's why it can't bounce back with high velocity.
The coefficient of the restitution which can be represented through the symbol e can be used to determine or describe the type of collision that can occur between the two moving bodies with the same mass or different velocities. It can simply defined through the given equation that can be written below:
e = relative velocity of seperationrelative velocity of approach
Or,
e = v2f- v1fv1i - v2i
This equation can be used for the determination of the type of collision between the objects such as;
To understand the conservation of energy or understand the concept of the interaction and the transferring of energy into another form, the concepts of elastic and inelastic collision are crucial to understanding because without understanding these concepts it can't be possible to understand the motion of two moving bodies efficiently. In the ideal system, both kinetic and momentum can be conserved but in reality, it can't be possible. In the real world mostly and commonly only partially inelastic collisions reoccurred. By understanding and reading the concept of collisions with their definitions, types, representations, derivations, and examples the reader can determine the types of collisions that can occur in their surroundings.