Momentum Formulas for K-12 Students

A comprehensive guide to understanding momentum across grade levels

Elementary School (K-5)

Introduction to Momentum

Momentum is what makes moving objects hard to stop. The heavier an object is and the faster it moves, the more momentum it has.

Simple Definition:

Momentum = Mass × Speed

🐢

Low Momentum

(Low Mass, Low Speed)

🚗

High Momentum

(High Mass, High Speed)

Example:

Which has more momentum: a heavy truck moving slowly or a light bicycle moving quickly?

It depends on how heavy and how fast each one is! The momentum depends on both mass and speed together.

Middle School (6-8)

Linear Momentum Formula

Basic Momentum Formula:

p = m × v

p = momentum (kg·m/s)
m = mass (kg)
v = velocity (m/s)

Momentum is a vector quantity, which means it has both magnitude and direction!

Example Problem:

Calculate the momentum of a 50 kg student running at 3 m/s.

Solution:

p = m × v = 50 kg × 3 m/s = 150 kg·m/s

Direction Matters!

When an object moves to the right, we can say its momentum is positive (+).

When an object moves to the left, we can say its momentum is negative (-).

🚲

→

Positive Momentum

🚲

←

Negative Momentum

High School (9-10)

Conservation of Momentum

In a closed system with no external forces, the total momentum before an event equals the total momentum after the event.

Law of Conservation of Momentum:

p_{total before} = p_{total after}

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

m₁ and m₂ = masses of objects
v_1i and v_2i = initial velocities
v_1f and v_2f = final velocities

Example: Collision of Two Carts

A 2 kg cart moving at 3 m/s collides with a stationary 1 kg cart. After the collision, they stick together. What is their final velocity?

Solution:

Initial momentum = Final momentum

m₁v_1i + m₂v_2i = (m₁ + m₂)v_f

(2 kg)(3 m/s) + (1 kg)(0 m/s) = (2 kg + 1 kg)v_f

6 kg·m/s = 3 kg × v_f

v_f = 2 m/s

Impulse-Momentum Theorem

Impulse is the product of force and time, which equals the change in momentum.

Impulse Formula:

J = F × Δt = Δp = m × Δv

J = impulse (N·s)
F = force (N)
Δt = time interval (s)
Δp = change in momentum (kg·m/s)
Δv = change in velocity (m/s)

Example: Catching a Ball

A 0.5 kg ball is moving at 10 m/s. If you catch it and bring it to rest in 0.1 seconds, what force did you apply?

Solution:

Δp = m × Δv = 0.5 kg × (0 - 10) m/s = -5 kg·m/s

F = Δp / Δt = -5 kg·m/s / 0.1 s = -50 N

The negative sign indicates the force is in the opposite direction to the initial velocity.

Applications of Impulse:

Air Bags

Increase time of impact to reduce force

Sports Equipment

Padding extends collision time to reduce force

Advanced High School (11-12)

Types of Collisions

Elastic Collisions:

Both momentum and kinetic energy are conserved.

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

\(\frac{1}{2}\)m₁v_1i² + \(\frac{1}{2}\)m₂v_2i² = \(\frac{1}{2}\)m₁v_1f² + \(\frac{1}{2}\)m₂v_2f²

Special Case: One-Dimensional Elastic Collision Formulas

v_1f = \(\frac{m_1 - m_2}{m_1 + m_2}\)v_1i + \(\frac{2m_2}{m_1 + m_2}\)v_2i

v_2f = \(\frac{2m_1}{m_1 + m_2}\)v_1i + \(\frac{m_2 - m_1}{m_1 + m_2}\)v_2i

Inelastic Collisions:

Momentum is conserved, but kinetic energy is not conserved (some is converted to heat, sound, etc.).

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

Perfectly Inelastic Collisions:

Objects stick together after collision. Momentum is conserved, but maximum kinetic energy is lost.

m₁v_1i + m₂v_2i = (m₁ + m₂)v_f

Example: Two-Dimensional Collision

A 3 kg object moving at 4 m/s east collides with a 2 kg object moving at 3 m/s north. If they stick together, what is their final velocity (magnitude and direction)?

Conservation of momentum in x-direction:

p_x,initial = p_x,final

(3 kg)(4 m/s) + (2 kg)(0 m/s) = (5 kg)(v_x)

v_x = 2.4 m/s east

Conservation of momentum in y-direction:

p_y,initial = p_y,final

(3 kg)(0 m/s) + (2 kg)(3 m/s) = (5 kg)(v_y)

v_y = 1.2 m/s north

Final velocity magnitude:

v = √(v_x² + v_y²) = √((2.4 m/s)² + (1.2 m/s)²) = 2.68 m/s

Direction (angle from east):

θ = tan^-1(v_y/v_x) = tan^-1(1.2/2.4) = 26.6° north of east

Angular Momentum

Angular Momentum Formula:

L = r × p = r × mv = Iω

L = angular momentum (kg·m²/s)
r = position vector (m)
p = linear momentum (kg·m/s)
I = moment of inertia (kg·m²)
ω = angular velocity (rad/s)

For a point mass moving in a circle:

L = mr²ω

Angular momentum is conserved when no external torque acts on a system.

L_initial = L_final

I₁ω₁ = I₂ω₂

Example: Figure Skater's Spin

A figure skater spinning with arms extended has an angular velocity of 2 rad/s and a moment of inertia of 4 kg·m². When she pulls in her arms, her moment of inertia decreases to 1 kg·m². What is her new angular velocity?

Solution:

Conservation of angular momentum: I₁ω₁ = I₂ω₂

(4 kg·m²)(2 rad/s) = (1 kg·m²)(ω₂)

ω₂ = 8 rad/s

The skater spins 4 times faster when she pulls in her arms!

Practical Applications

Real-World Applications of Momentum

Sports

Billiards and pool (collisions)
Baseball and cricket (impulse)
Figure skating (angular momentum)
Gymnastics (conservation of angular momentum)

Transportation

Car safety (airbags, crumple zones)
Rocket propulsion (conservation of momentum)
Train collisions (inelastic collisions)
Aircraft carrier landing systems (impulse)

Physics and Engineering

Space exploration (conservation of momentum)
Ballistic pendulum (conservation of momentum)
Particle colliders (elastic collisions)
Recoil in firearms (conservation of momentum)

Everyday Life

Hammering a nail (impulse)
Walking and running (impulse)
Playground seesaw (angular momentum)
Amusement park rides (conservation of momentum)

Quick Reference Table

Formula Name	Equation	Variables	Grade Level
Linear Momentum	p = mv	p = momentum, m = mass, v = velocity	6-12
Conservation of Momentum	p_before = p_after	p = total momentum	9-12
Impulse	J = F × Δt = Δp	J = impulse, F = force, Δt = time interval, Δp = momentum change	9-12
Perfectly Inelastic Collision	m₁v_1i + m₂v_2i = (m₁ + m₂)v_f	m = mass, v = velocity, i = initial, f = final	10-12
Angular Momentum	L = r × p = Iω	L = angular momentum, r = position vector, p = linear momentum, I = moment of inertia, ω = angular velocity	11-12