Linear Momentum

claimed by joshua kim fall 2025

Linear Momentum

It is a vector quantity that describes an object's motion by combining its mass and velocity. Because mass is a scalar while velocity is a vector, momentum always points in the same directions as an object's motion. Momentum is typically represented by the symbol [math]\displaystyle{ \vec{p} }[/math], and its SI unit is the kilogram meter per second (kg*m/s). Momentum is sometimes referred to simply as "momentum", and the plural form may appear as either "momenta" or "momentums".

The Main Idea

Momentum reflects how difficult it is to stop or change the motion of an object. Increasing either the mass or the velocity of an object increases its momentum. Momentum plays a central role in understanding collision, motion in systems of particles, and many other physical processes because it is often conserved even when forces are complicated or short-lived.

A Mathematical Model

For a single particle, linear momentum is defined by [math]\displaystyle{ \vec{p} = m\vec{v} }[/math], where [math]\displaystyle{ m }[/math] is the mass and [math]\displaystyle{ \vec{v} }[/math] is the velocity. This formula works well for everyday speeds, but for motion approaching the speed of light, the definition must be updated using relativistic momentum.

For systems of multiple particles, the total momentum is the vector sum of the momenta of all particles: [math]\displaystyle{ \vec{p}_{\text{total}} = \sum m_i\vec{v}_i }[/math]. This allows complex systems - such as a rolling disk with many moving pieces - to be treated as if all their mass were concentrated at a single point.

Single Particles

The momentum of a particle is defined as follows:

[math]\displaystyle{ \vec{p} = m\vec{v} }[/math]

where [math]\displaystyle{ \vec{p} }[/math] is the particle's linear momentum, [math]\displaystyle{ m }[/math] is the particle's mass, and [math]\displaystyle{ \vec{v} }[/math] is the particle's velocity. This formula accurately describes the momentum of particles at everyday speeds, but for particles traveling near the speed of light, the formula for Relativistic Momentum must be used for concepts such as the momentum principle to remain true.

Multiple Particles

The total momentum of a system of particles is defined as the vector sum of the momenta of the particles that comprise the system:

[math]\displaystyle{ \vec{p}_{system} = \sum_i \vec{p}_i }[/math]

Although the proof does not appear on this page, it can be shown that the total momentum of a system of particles is equal to the total mass of the system times the velocity of its Center of Mass:

[math]\displaystyle{ \vec{p}_{system} = M_{tot}\vec{v}_{COM} }[/math]

This formula makes it significantly easier to calculate the momentum of certain objects. For example, consider a disk rolling along the ground. The disk is comprised of an infinite number of infinitely small "mass elements," all of which have different momenta; the ones at the bottom of the disk are hardly moving while the ones at the top are moving very quickly. Without the formula above, one would have to use an integral to add the momenta of all of the mass elements to find the total momentum of the disk. However, the formula above tells us that we can simply multiply the mass of the disk by the velocity of its center of mass, which is in this case the geometric center of the disk. This reveals that the fact that the disk is rotating does not affect its momentum.

Impulse

Impulse measures the change in momentum during a time interval. It is defined as the integral of the force over time, and reduces to [math]\displaystyle{ \vec{J} = \vec{F}\Delta t }[/math]. This relationship is extremely useful in analyzing collisions, sports physics, and any situation where short-duration forces cause a measurable change in motion.

In Relation to Other Physics Topics

When no net external force acts on a particle, its momentum remains constant (Newton's First Law). When a force does act, the rate at which the momentum changes equals the net force (Newton's Second Law). In situations where an impulse is applied, the momentum changes by an amount equal to that impulse. For systems in which the total external force is zero, momentum is conserved even if internal forces act between particles.

=Momentum in Rocket Propulsion

Rockets operate by expelling high-speed exhaust gases backward, which produces a forward momentum change in the rocket. Even in deep space, where there is "nothing to push against", momentum conservation ensures that ejecting mass at high velocity generates thrust. This highlights why rockets require large amounts of fuel: without a constant supply of propellant, the rocket cannot continue changing its momentum.

A Common Misconception: Newton's Cradle

Although Newton's cradle is often use to demonstrate conservation of momentum, the device actually relies on both momentum and energy conservation to produce its familiar motion. If momentum alone were conserved, the balls would not swing back with the same amplitude. The cradle highlights how both conservation laws work together in elastic collisions.

A Computational Model

In computational simulations of particles using Iterative Prediction, a momentum vector variable is assigned to each particle. Such simulations usually in "time steps," or iterations of a loop representing a short time interval. In each time step, the particles' momenta are updated based on the forces acting on it. Then their velocities are calculated by dividing each particle's momentum by its mass. Finally, the velocities are used to update the positions of the particles. Below is an example of such a simulation:

This vPython simulation shows a cart (represented by a rectangle) whose motion is affected by a gust of wind applying a constant force.

https://trinket.io/glowscript/ce43925647

For more information, see Iterative Prediction.

Examples

1. (Simple)

Find the momentum of a ball that has a mass of 70kg and is moving at <1,2,3> m/s.

Steps:

1. Initialize Variables

  m = 70 kg

  v = <1,2,3> m/s

2. Momentum Principle

  p = m * v

  = 70 kg * <1,2,3> m/s

  = <70,140,210> kg m/s

Therfore, the momentum of the ball is <70,140,210> kg m/s.

2. (Middling)

A car has 20,000 N of momentum. How would the momentum of the car change if:

a) the car slowed to half of its speed?

b) the car completely stopped?

c) the car gained its original weight in luggage?

Steps:

1. Initialize variables:

  p = 20,000 N
  p_n = ? (new momentum)

a. Half speed:

  p = m * v = 20,000 N   If velocity was halved:

  v = 1/2 v   

  p_n = m * 1/2 v

  = 1/2 * m * v

  = since m * v = 20,000 N, then

  p_n = 10,000 N

  In other words, if the velocity was halved, then momentum would 
  correspondingly be halved.

b. Car completely stopped:

  If the car was completely stopped, then its velocity would be 0.

  Since momentum is m * v, then the new momentum would be 0 as well.

c. Gained original weight in luggage:

  Double weight = 2m

  p_n = 2m * v

  = 2 * m * v

  = 40,000 N

  The car's new momentum would be doubled if the mass was doubled.

3. (Difficult)

You and your friends are watching NBA highlights at home and want to practice your physics. You notice at the beginning of a clip a basketball ball is rolling down the court at 23.5 m/s to the right. At the end, it is rolling at 3.8 m/s in the same direction. The commentator tells you that the change in its momentum is 17.24 kg m/s to the left. Curious at how many basketballs you can carry, you want to find the mass of the ball.

Steps:

  1. Initialize Variables:
  
  v_i = 23.5 m/s   v_f = 3.8 m/s

  p = -17.24 kg m/s (negative because direction is left)

  2. Plug in values and solve for m:

  -17.24 = m(3.8-23.5)

  m = -17.24/-19.7
  
  = 0.8756 kg (approximately)

4. (Difficult)

A system is comprised of two particles. One particle has a mass of 6kg and is travelling in a direction 30 degrees east of north at a speed of 8m/s. The total momentum of the system is 3kg*m/s west. The second particle has a mass of 3kg. What is the second particle's velocity? Give your answer in terms of a north-south component and an east-west component.

Steps:

  1. Initialize Variables:

  m_1 = 6 kg   v_1 = 8 m/s 30 degrees east of north

  m_2 = 3 kg   p_total = 3 kg m/s west

  2. Break Down Velocities into Components for Particle 1:

  North Component: v_1N = v_1 * cos(30) = 6.9282 m/s

  Eastward Component: v_1E = v1 * sin(30) = 4 m/s

  3. Calculate Momentum Components for Particle 1:

  North Momentum: p_1N = m_1 * v_1N = 6 * 6.9282 = 41.5692 kg m/s

  East Momentum: p_1E = m_1 * v_1E = 6 * 4 = 24 kg m/s

  4. Express Total Momentum Components:

  North total momentum:
  
  p_total,N = 0 kg m/s (no north-south component in total momentum)

  Eastward total momentum (negative because west is negative east):
  
  p_total,E = -3 kg m/s 

  5. Conservation of Momentum: 
  
  p_total,E = p_1E + p_2E

  Particle 2 North Momentum: 0 = 41.5692 + p_2N, p_2N = -41.5692 kg m/s

  Particle 2 East Momentum: -3 = 24 + p_2E, p_2E = -27 kg m/s

  6. Calculate Velocity Components:

  North Velocity: v_2N = p_2N / m_2 = -13.8564 m/s

  East Velocity: v_2E = p_2E / m_2 = -9 m/s

  Therefore, the second particle's velocity components are:

  13.86 m/s to the South

  9.00 m/s to the West

Connectedness

Scenario: runaway vehicle

Imagine that you are standing at the bottom of a hill when a runaway vehicle comes careening down. If it is a bicycle, it would be much easier to stop than if it were a truck moving at the same speed. One explanation for this is that the truck would have a greater mass and therefore a greater momentum. In order to be brought to rest, the truck must therefore experience a large change in momentum, which means a large impulse must be exerted on it.

History

The oldest known attempt at quantifying motion using both an object's speed and mass was that of René Descartes (1596–1650) (source). John Wallis (1616 – 1703) was the first to use the phrase momentum, and to define it as the product of mass and velocity (rather than speed). He recognized that this notion of momentum is conserved in closed systems (source), a concept confirmed by experiments performed by Christian Huygens (1629-1695) (source). Finally, Isaac Newton (1643-1727) wrote his famous three laws relating momentum to force in his book Principia Mathematica in 1687. These offered a theoretical explanation for conservation of momentum. These foundations were so logically sound and experimentally observable that they would go unquestioned for centuries, until Albert Einstein (1879-1955) had to adjust the formula for momentum to maintain its accuracy for objects moving at relativistic speeds (see relativistic momentum).

See also

• Mass
• Velocity
• Vectors
• Newton's Second Law: the Momentum Principle
• Impulse and Momentum
• Conservation of Momentum

Further reading

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 1). Raleigh, North Carolina: Wiley.

External links

1. https://www.khanacademy.org/science/physics/linear-momentum/momentum-tutorial/v/introduction-to-momentum

References

• http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html
• http://study.com/academy/lesson/linear-momentum-definition-equation-and-examples.html
• https://science.jrank.org/pages/4419/Momentum.html
• https://www.famousscientists.org/john-wallis/
• https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html