Introduction to Momentum
Simply, momentum is a mass in motion. All the moving objects like a moving person, a moving car, a football in motion or even a propelled rocket exhibit a common property called the ‘momentum’. It gives the strength to the motion of any object possessing a certain mass. Momentum is the property that mathematically speaks how a moving object can be stopped or the direction of that object can be changed. Momentum is where all the mechanics are concentrated. In the present day, new branches of physics like quantum mechanics and the general theory of relativity are also based on momentum. We can say, laws of motion are derived from the behavior of momentum of any moving object.
Mathematically, momentum can be defined as the product of the mass of that moving object and its velocity. It depends on the mass and velocity of the moving object. Therefore a body with faster motion can also create more, damage in collisions as the velocity is significantly increased..
Momentum is the root of physics where the force, motion, and mass are concentrated. It also helps how objects interact with each other. Moreover, it plays a vital role in designing projects that are to be brought in motion.
Concept and History of Momentum
Momentum can also be defined as the quantity of motion carried by a moving body. It has a direct relation with the mass and the velocity of that body. Therefore, a body with greater mass and greater velocity has greater momentum.
Conversely, if an object is at rest, its momentum is zero as the body acquires no motion and hence no velocity. The same object on starting motion automatically gains momentum.
The word “momentum” is derived from the Latin word movimentum, which means “movement.The motion of a body was first talked about by Aristotle and stated that for a body to remain in motion, it should continuously be supplied by an external force. This statement dominated physics for many years. Later, Galileo Galilei studied moving objects during the 1600s and gave a conclusion that a body can continue in its motion until it is acted upon by some external forces. He brought the concept of inertia which was closer to the concept of momentum.
In the 17th century Sir Isaac Newton came up with the revolutionary concept of motion which is published in Principia Mathematica. He studied Galileo’s idea and derived three laws of motion. His first law was concentrated on the inertia of rest and motion. The second law of motion is the law where the term “momentum’ was described for the first time. He gave the mathematical definition of momentum (p = mv) and stated that the rate of change of momentum of a body is directly proportional to the force applied to it. He also gave the conservation law for momentum saying that for an isolated system where no force is acting, the total momentum of the system remains constant. This became the point for the evolution of classical mechanics.
Momentum is a vector quantity, having both magnitude and direction. The direction of momentum is the same as the direction of the velocity of that object.
For example: a bike moving towards east has its momentum directed toward the east. If the bike now changes its direction towards west, the momentum also changes to the west as that body.
Mathematical Expression and Units of Momentum
There is a simple formula to calculate momentum. If a body has a mass ‘m’ which is moving with the velocity ‘v’, its momentum is indicated by p and given as:
p = mass (m)×velocity (v) [Equation 1]
Hence, the above equation shows that momentum depends directly on the mass and velocity of the moving object.
Its SI unit is kilogram meter per second (kg·m/s) or Newton-second (N·s).
For example:
If a car of mass 1000 kg moves with a velocity of 80 m/s, its momentum is calculated as:
p = 1000×80 = 80,000 kg·m/s
This means the car acquire a momentum of 80,000 kg·m/s in the direction of its motion.
Dimensional Formula
[Momentum]=[M^1] [L^1T^−1] = [M^1L^1T^-1]
This shows momentum depends on mass (M), length (L), and time (T).
Types of Momentum
There are two types of momentum known as: linear momentum and angular momentum.
Linear momentum:
It is the most common type of momentum studied in physics. Linear momentum means the motion that appears in a straight line. It describes the motion of the objects along a particular direction with a certain speed. As momentum is a vector quantity, the arrow sign over it shows the direction of movement. Some of the examples of linear momentum may be, a football kicked by a player, a moving car, a running man, a bullet fired from a gun along a straight line etc.
Here, the condition applied for the linear momentum is that the motion is in a straight line, and the direction of momentum is same as that of the direction of velocity. If two objects move in opposite directions, as per the sign convention of the vector, the momenta will have opposite signs. For example, if one body is moving towards east and is considered to have positive momentum, then the body moving towards west must be regarded to have negative momentum.
Angular momentum:
Angular momentum comes into action when the objects are moving along a circular path. It is calculated as the product of moment of inertia of the object and its angular velocity.
L= I × ω
where,
- I is the moment of inertia and
- ω is the angular velocity.
It is also a vector quantity and the total angular momentum of a closed system is conserved. For example, flywheels, spinning figure skaters, gymnasts, rotating fans, centrifuges etc.
Conservation of Momentum
The ” Law of Conservation of Momentum” has a special place in physics. It states that for an isolated system where no forces like friction, gravity or air resistance are acting, the total momentum of the system remains constant.
Mathematically, if two bodies of masses m1 and m2 hit each other with initial velocities u1 and u2, their final velocities will be v1 and v2. Now, their momentum before and after the collision takes the form:
m1u1+m2u2=m1v1+m2v2 [Equation 2]
For example, a moving truck collides with a car at rest. Then, after collision either both may move together or both may move independently with different final velocities. However, the total momentum before and after their collision will be the same.
Thus, when two bodies exert forces on each other, their forces become equal and opposite. This results in the conservation of momentum.
Impulse and Change in Momentum
Sometimes, forces do not act longer and are of very short duration. For example, catching a ball, when a bat hits a ball, a hammer striking a nail etc. This force is known as impulse and is the product of the force and the time interval during which it acts.
Impulse=Force×Time [Equation 3]
Impulse can also be defined in another way as “the change in momentum of an object”, which is mathematically given as:
J = Δp = m(v−u) [Equation 4]
Where,
- u = initial velocity
- v = final velocity
Its SI unit is also same as the momentum i.e. N·s (Newton-second).
For example, when a player catches a fast-moving ball, he lowers his hands. This increases the time of contact over which the ball’s momentum is brought to zero and hence the hands can feel less force.
Thus, impulse explains how increasing the time of contact can reduce the effect of a large force. Same principle used by engineers in safety devices like airbags and helmets.
Applications of Conservation of Momentum
There are huge applications of the law, some of them are given below:
- Rocket Propulsion
When a rocket is fired, it throws gases backward at high speed.The momentum of the rocket is equal in magnitude but opposite in direction to the momentum of the gases. This makes it able to move in space, with constant momentum.
- Recoil of a Gun
When a bullet is fired from a gun, the bullet moves with forward momentum and the gun moves backward or recoils. This makes the momentum equal and opposite.
- Colliding objects
In accidents or any collision involving events, the conservation of momentum is applied and is used to investigate the velocities of those bodies before and after the collision.
- Sports
In football, cricket, billiards, or baseball, momentum plays a big role. Players use ideas to control the change in momentum, increase the contact time etc. to achieve better accuracy and reduce the impact of force.
Elastic and Inelastic Collisions
Collisions are the best and direct examples based on the law of conservation of momentum. Two types of collisions are: elastic and inelastic collisions.
- Elastic Collision
Here, both the momentum and kinetic energy are conserved.
For example, collision between gas molecules in an ideal gas, collision of small steel balls etc..
Mathematically:
m1u1+m2u2=m1v1+m2v2 [Equation 5]
And
½ m1u1^2+½ m2u2^2=½ m1v1^2+½ m2v2^2 [Equation 6]
- Inelastic Collision
In an inelastic collision, only momentum is conserved but kinetic energy is lost. Some energy is lost as sound, heat, or deformation of the objects. For example, when a car hits a bicycle, momentum of the striking objects remains conserved but their kinetic energy changes and either they both start moving combinedly or with some decreased kinetic energy after the collision.
- Perfectly Inelastic Collision
In this case the two objects meeting collision will stick together and move as one body. This is the peak condition of an inelastic collision. For example, clay sticks on the floor after colliding with it.
Momentum in Two and Three Dimensions
The above described cases are the normal cases of collision which are along one-direction. There are also some practical situations which are in two or three dimensions. For example, a spherical ball moving with a certain velocity vector, strikes with a vertical wall at a certain angle. In such conditions, momentum has to be calculated as of two or three dimensions.
Momentum being a vector quantity, it can be resolved into its components along the x-, y-, and z-axes. The law of conservation of momentum can be applied independently along each direction.
For example, two balls collide on a table. Before the collision, each ball has its own velocity in some direction. After collision, their velocities change. But if we consider momentum along the x-axis and y-axis separately, the total momentum along each axis remains the same.
- Along x-axis: m1u1x+m2u2x=m1v1x+m2v2x
- Along y-axis: m1u1y+m2u2y=m1v1y+m2v2y
In 3D motion, the momentum conservation follows, m1u1yz+m2u2z=m1v1yz+m2v2yz
This momentum analysis and calculation is very useful in studying projectile motion, explosions, and particle collisions.
Practical Examples and Importances
Momentum is a natural phenomenon for every moving object. So, it is a most common physical property observed in nature. Some familiar examples are given below:
- Walking and Running
When we walk or run, we apply a backward force on the ground. In return, the ground pushes us forward and makes us able to walk or run. Hence, this action and reaction forces provide momentum for us.
- Vehicle Safety Systems
Modern vehicles are designed with crumple zones and airbags. This increases the time of contact during the collision and hence reduces the impulse force, which is the rate of change in momentum. Hence, this reduces the severe harm on the passengers during crashes or accidents.
- Sports
Athletes, gymnasts, footballers, cricketers, shot put players and many others adjust their body movements and also make predictions and analyses for increasing or decreasing momentum. For example fielders in cricket lower their hands while catching a ball to reduce the contact time and hence reduce any possible injuries.
- Spacecrafts designs
Satellites and spacecraft work with small accelerators to control their momentum. Even a tiny release of gas in one direction can change the spacecraft’s speed in the opposite manner. Hence, tiny accelerators provide a precisely desired control of the spacecraft in space.
- Everyday Transportation
To reduce a vehicle’s momentum and make it stop safely, the brakes are designed to produce a force in the opposite direction of motion.
Conclusion
Momentum is a core of physics where all the major branches like classical mechanics, quantum mechanics, thermodynamics and general theory of relativity rely. Since every object in the earth or even in the universe is moving, the study of momentum can reveal certain details about the universe. The four types of forces: strong, nuclear, weak or gravitational forces, all have some impacts on momentum.
Practical life and our daily circumstances are also roaming around momentum. Thus, it is practically impossible to perform any motion without momentum. Conservation of momentum is applied to all the moving objects and hence it is like universal law for the collisions. Momentum calculation is also done in the engineering and physics lab to design new technologies of motion. A rocket and a spacecraft also start on the calculation of momentum. Hence, it is the simple but the most powerful topic in physics and engineering. The study of momentum is everlasting.
References
Salah, A. R. M. (2025). Conservation of Linear Momentum: Principles, Proofs, and Applications.
Galante, L., & Gnesi, I. (2020). Two-penny physics: Teaching 2D linear momentum conservation. American Journal of Physics, 88(4), 279-285.
Wilson, E. B. (1915). Linear Momentum, Kinetic Energy, and Angular Momentum. The American Mathematical Monthly, 22(6), 187-193.
https://en.wikipedia.org/wiki/Momentum
https://www.geeksforgeeks.org/physics/momentum-formula
