Hello friends, I hope you are all well and doing your best in your fields. Today we can discuss the fundamental concept of momentum which can play a very crucial role in physics. To understand the motion of the moving object, understanding the concept of momentum is essential. like the velocity, displacement, and momentum are also vector quantities because they can describe the both magnitude and direction of the moving body. Momentum is the product of the mass and the velocity of the objects so it is the vector quantity. The quantity of the motion can be determined through the momentum. Because when the rate of change of force that can be acted on the body is changed, momentum also changes because the rate of change of force is equal to the rate of change of momentum.
In some systems, momentum is conserved when external forces act on the system externally but when different forces act on the system then mostly momentum can't be conserved. Momentum can describe massive objects that can move with high velocity and move faster. Now we start our detailed discussion and explore the definition of momentum, mathematical representation, their formula for single moving particles or many-particles, their types, examples, significance, applications, and problems.
Concept of the momentum is fundamental but it is the study of the quantity of motion so they have a rich history the first scientist who discovered or understood the concept of momentum was Aristotle because he was the first who understand the motion of the moving bodies. After Aristotle, galileo researched and collected more deep quantitative information about the crucial concept of momentum. After these scientific efforts and with their information teh most famous scientist Issac Newton understood the new and modern concept about the motion of moving bodies with momentum and presented the new law of observation of momentum in which the momentum of the moving bodies in teh isolated systems always remains constant or conserved because in isolated systems no external forces acting on the moving particle or teh body.
The basic definition of momentum for a single moving particle is given there:
“ the product of the mass and the velocity of the moving object or body are termed as the momentum. Because in momentum we determine the quantity of the motion.”
Mathematically momentum can be represented as:
ρ = mv
There,
ρ represented the momentum of the moving body.
m represented the mass of the moving object.
v represented the velocity of the moving object.
Momentum is the product of the velocity and the mass so the unit of mass is kg (kilogram) and the unit of velocity is msec (msec-1) ( meter per second). Hence, the unit of momentum is kg msec-1 , newton second (N sec), or gram centimeter per second ( g cm sec-1).
Dimension for the unit of momentum is MLT-1.
Momeyum is the vector quantity so the direction of the momentum is the same as the direction of the velocity of the moving object.
In momentum, the magnitude of the moving body is its mass. For instance, if the 1kg mass of the body moves in the road in the south direction then its magnitude is 1kg and its direction is south so momentum is a vector quantity so they can provide information about both magnitude and direction.
The total momentum for different particles that can be moved in a system is the sum of the individual moving particle momentum. let us consider the two moving particles with mass m1 or m2 and moved with the velocity v1 and v2 then there total momentum is represented as:
ρ = ρ1 + ρ2
Or,
ρ1 = m1v1
ρ2 = m2v2
So,
ρ = m1v1 + m2v2
If the system has many different particles or more than two particles then we can find their momentum by using the given formula:
ρ = i mivi
Momentum is the product of the velocity and the mass so the unit of mass is kg (kilogram) and the unit of velocity is msec (msec-1) ( meter per second). Hence, the unit of momentum is kg msec-1 , newton second (N sec), or gram centimeter per second ( g cm sec-1).
As we know,
1N = kg ms-2
So,
N s = kg ms2 s
N s = kg ms
N s = kg m s-1
Hence, it proved that kg msec-1 = N s
Dimension for the unit of momentum is MLT-1.
Momeyum is the vector quantity so the direction of the momentum is the same as the direction of the velocity of the moving object.
In momentum, the magnitude of the moving body is its mass. For instance, if the 1kg mass of the body moves in the road in the south direction then its magnitude is 1kg and its direction is south so momentum is a vector quantity so they can provide information about both magnitude and direction.
When the constant force can be applied to the body, but the force can be applied on the body with some time of interval but when the force and time interval change then the momentum of the body can also be changed and mathematically it is written as:
Δρ = FΔt
There,
Δρ = change in momentum of the moving body.
F = constant force that can be applied to the moving body.
Δt = time interval when the constant force is applied to the moving body.
Let's suppose that the body can be moved with the mass m and with their initial velocity vi. during their motion, the force F can be applied on the body constantly with the time interval t, and the moving body can change its velocity in the final point which is represented as vf. now during the motion of the body acceleration can also be produced and mathematically the acceleration of the moving body can be represented as:
a = vf- vit
Then, according to Newton's second law of motion,
F = ma
There, F indicates the force that can be applied on the moving body, m indicates the mass of the moving object and a indicates the acceleration of the moving body.
Then put the equation for acceleration in F = ma equation and write as:
F = m vf- vit
Then,
F = mvf-mvit
Now, according to the given equation
mvi = initial momentum for the moving body.
mvf = final momentum for the moving body.
According to the second law of motion, momentum can be stated as:
“The change in the momentum with the interval of time is always equal to the force that can be applied to the moving body.” momentum according to the second law of motion can easily apply to those moving bodies where their mass can be changed.
Some properties of the momentum are given there:
Vector quantity: momentum is the vector quantity because it can provide information about the direction and the magnitude of the moving object.
Mass and velocity: The mass and velocity of the moving object directly depend on the momentum because according to the equation ρ = mv, when the mass and the velocity of the moving object are greater then teh momentum of the body is also greater. The fast-moving object with a heavy mass has the greater momentum.
Conserved quantity: in the isolated system in which no external forces can act on the body their momentum can be conserved because they are moving in a closed system but when the system is not isolated and many forces act on them then their momentum is not conserved. The system in which the momentum is conserved is termed the law of conservation of momentum.
The conservation of momentum is also the fundamental concept of momentum. Momentum always remains constant or conserved in teh isolated system or the closed system where no external force can act on it. The law of conservation of momentum is mostly used to determine the velocity and the momentum after a collision between the two different moving particles which have different velocities but have the same masses. their mathematical representation and their formula are given there:
Let's suppose the two moving particles have the same masses but have different velocities before and after the collision but their momentum is conserved because in both moving bodies, no external forces are acted and it can be written as:
m1v1i + m2v2i = m1v1f + m2v2f
There, m1, m2 represented teh mass of the two different moving objects and v1i , v2i represented the initial velocity of the two moving objects and v1f , v2f represented the final velocity of the two moving objects.
But if many objects can be moved in the isolated system then their momentum can also be conserved and determined through the formula that is given there:
ρinitial = ρfinal
m1v1i + m2v2i …… mnivni= m1v1f + m2v2f…… mnfvnf
In a collision, the momentum can be conserved. In types of collision, the momentum is always conserved like in the elastic collision and the inelastic collision their detail is given there:
Elastic collision is defined as:
“ when kinetic energy and momentum 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 representation:
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.
Inelastic collision is defined as:
“The kinetic energy and the momentum 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
Impulse can be defined as:
“ the cross product between the force and the time is termed as the impulse of force. In an impulse of force, a very large amount of force acts on the body but it can act on the body for a very short interval of time.
Impulse mathematically can be represented as:
I = F t
There,
I represented teh impulse of the force.
F represented the force that can be acted on the body
t represented the time interval in which the force can be acted on it.
The impulse of force is the product of the force and the time so the unit of F is and the unit of time is sec so their unit is N sec or kg msec-1.
Dimension for the unit of the impulse of force is MLT-1.
The relationship between the impulse of force and the momentum can be shown by the given derivation:
According to the second law of motion,
F = mvf-mvit
Now by using the formula of the impulse of force,
I = F t
Now put the value F in the formula of the impulse of force as
I = mvf-mvit t
Then,
I = mvf- mvi
By this equation, it is proved that the impulse of the force is equal to the momentum as,
Impulse of force = momentum
I = ρ
Now according to this equation impulse can also be defined as the:
“The change in the momentum due to the impulsive forces is termed as the impulse.”
Impulse can also be mathematically represented as:
ΔJ = Δρ = FΔt
There,
ΔJ represented the impulse
Δρ represented the change in the momentum
Δt represented the change in the time
Impulsive forces can be defined as:
“The force that can be acted on the body in a short interval of time is termed as the impulsive of forces.”
The force that can be acted on the body for a short period, sometimes force can act on the body for a very short interval of time but the force thrust is very high so that's why the great force acts on the body for short intervals called impulse. For instance, when the cricketer plays a match then the ball that can be thrown can hit the ball with great force so the force can act for a short interval of time with impulsive forces termed as an impulse.
Types of momentum:
There are two major types of momentum which are given:
Angular momentum
Linear momentum
Linear momentum can be defined as:
“The body that can be moved in a straight line, then their momentum is termed as the linear momentum.” linear momentum is the product of the mass and velocity.
Mathematically linear momentum can be represented as:
ρ = mv
There,
ρ represented the momentum of the moving body.
m represented the mass of the moving object.
v represented the velocity of the moving object.
Linear Momentum is the product of the velocity and the mass so the unit of mass is kg (kilogram) and the unit of velocity is msec (msec-1) ( meter per second). Hence, the unit of momentum is kg msec-1 , newton second (N sec), or gram centimeter per second ( g cm sec-1).
Dimension for the unit of linear momentum is MLT-1.
Angular momentum is the momentum that can be produced by the body during the rotational or circular motion of the body. However, the angular momentum of the rotational moving body is directly dependent upon the inertia of the body and also it depends upon the angular velocity of the body through which the moving body can be moved.
L = r ρ
There,
L represented the angular momentum.
r represented the position vector
ρ represented the momentum of the moving body.
In quantum mechanics, the concept of momentum is fundamental and observable through the momentum that can be operated during their wave function. Different scientist can present their information and describe the momentum concept or measurement in quantum mechanics but the principle of uncertainty that can be presented by Heisenberg describes that the momentum that can be measured can't be attained or achieved simultaneously. The equation or derivation that can be represented by these statements is given there:
Δx Δρ ħ2
There,
Δx represented the uncertainty in the position.
Δρ represented the change in momentum
ħ represented the Planck constant.
The concept of momentum is fundamental and crucial to understanding because relativity at high velocity can be determined or modified by the modern concept of momentum. So the equation that can be determined is the relativistic momentum of the moving object is given there:
ρ = y mv
ρ is the relativistic momentum
y Lorentz factor
m represented the mass of the body
v = represented the velocity of the moving body
Or the Lorentz factor can be defined or written as:
y = 11- v2c2
there, v represented the velocity of the moving body, and c represented the speed of light. and the relationship of momentum with velocity, mass, and speed of light can be shown through the equations that can be written above.
The angular momentum and the rotational motion are the same because in the rotational motion, the angular momentum can be produced and teh angular momentum directly depends upon the inertia of the moving object and also depends upon the angular velocity through which the body can be moved. Mathematically the rotational motion of the angular momentum can be represented as:
L = I ω
There,
L represents teh angular momentum or the momentum for the rotational motion
I represented the inertia of the moving body.
ω represented the angular velocity through which the body can be moved in the circular or the rotatory path.
The momentum of the moving bodies can also be studied or determined through experimental studies. In experimental studies, we can use different tools or instruments like high-speed cameras, and different types of tracking software that are used to measure or understand the velocities of the moving bodies before or after the collision. Through the experimental studies we can understand the theories and the formula that are used for measuring the momentum of the moving body. Through experimental studies, we can also understand the transformation of energy through another type of energy.
The advanced topics in which the momentum plays a crucial role and effect are given there:
Magnus effect
Air resistance and drag force
The baseball or the golf balls can spin with the spin effect and the projectile that can be formed by the baseball or golf is created due to the Magnus effect. The ball when spinning force can act on it but it acts on the ball in the perpendicular direction of the motion in which the body can be moved. When the great force acts on the ball then it can follow the curve which can be shown by the projection of flight and the Magnus effect.
The air resistance and the drag force can affect the momentum. The drag force can directly affect when the body can do projection because this force is equal to the square of the projectile velocity but this force can move to the opposite side in which the body can be moved. Due to the air resistance and the drag force the height, projectile, and range of projection and velocity can be reduced which makes the path of motion complex for the moving body.
Some applications of momentum in detail are given there:
Spacecraft navigation
Vehicle safety
Subatomic particle
sports
The spacecraft can maneuver due to the conservation of momentum in the space. In the spacecraft due to the conservation of momentum, the fuel or the gas can be expelled in one direction and the spacecraft moves opposite direction it can change its direction due to the momentum. It is not only used in the spacecraft this process or principle can also be used in the rocket propulsion.
The concept of momentum and the relationship of momentum with impulse can used in the safety of vehicles because their designing engineers can design seat belts, crumple zones, and many different parts according to their fundamental concept. Using these advanced features in the vehicle preserves or extends the life after the collision and reduces the risk of injuries due to the collision.
In the field of physics where we can discuss subatomic particles, we can understand the collision of the particles efficiently. Momentum also helps to understand the motion of the moving particles. By understanding the fundamental concept of momentum and their law of conservation the behavior of the particles can also be understood efficiently.
In sports, momentum can play a very fundamental role because it can help the athletes improve their control of games and also help to enhance their performances and improve their strategies. For instance, when the cricketers play the cricket game they can hit the ball with the greatest force and show impulse of force and also the relationship with momentum.
In modern physics or quantum physics, classical mechanics the concept of momentum is a cornerstone and crucial to understanding. because by understanding the momentum we can easily understand the motion of moving objects. In the modern physical world, the concept of momentum and the law of conservation of momentum can play a very important role. By exploring the details of the momentum through their experiential verification we can observe the momentum in our daily life. After understanding the concept of momentum the interactions and the collisions that can occur between the particles can also be understood. after reading this article the reader can understand the details of momentum and also collision, their types, and the law of conservation of momentum efficiently.
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