Equations of motion

Math helps us see how things move.

It shows where things go. It tells us how fast they move. This helps us know where they will be.

Scientists use special rules to track things. These rules use time to show motion. Some rules look at forces like pushes. Other rules only look at where things are. This helps us understand the world. It is like a map for moving things.

How do things move?

Scientists use math to answer this. They use equations of motion. These are rules that describe how things move over time.

There are two main ways to study motion. One way is called kinematics. This is a simpler way. It only looks at where an object is and how fast it goes. It does not look at why it moves. The other way is called dynamics. This is more complex. It looks at forces and power to explain the motion.

When an object moves with constant acceleration, we use the SUVAT equations. Acceleration is how much speed changes every second. These equations use five main parts. They use displacement, which is the change in position. They also use initial velocity, which is the starting speed. We also use final velocity, initial acceleration, and time.

Long ago, thinkers like Galileo studied these ideas. He studied how objects fall and how they fly through the air. Later, Isaac Newton helped make these rules even better. Today, these equations help us understand everything from small particles to big stars.

Equations of motion are special math rules. They describe how a physical system moves over time. Scientists use these rules to understand the behavior of objects. These equations use different variables to show movement. Most often, they use space and time to track an object. Some equations even use momentum, which is a part of how things move. These functions work in different ways depending on the type of science being studied.

There are two main ways to describe motion. The first way is called kinematics. Kinematics is a simpler method. It only looks at variables like position and time. It does not look at the forces that cause the movement. The second way is called dynamics. Dynamics is more general and complex. It takes into account forces, energy, and momentum. When acceleration is constant, we use the SUVAT equations. These equations use displacement, initial velocity, final velocity, acceleration, and time.

Many thinkers helped develop these ideas over thousands of years. In ancient times, people used math to predict eclipses. However, they did not write down equations of motion for a long time. In the thirteenth century, scholars at Oxford and Paris studied physics. Thomas Bradwardine and Nicholas Oresme worked on these ideas. Later, a Spanish thinker named Domingo de Soto wrote about how motion relates to time. He correctly noted that acceleration can be negative when something moves upward.

Galileo Galilei made huge progress in this field. He showed that the path of a flying object is a parabola. Galileo also studied a swinging pendulum in a cathedral in Pisa. He discovered that the time for a swing stays the same even as it slows down. He measured momentum by multiplying velocity and weight. Later, scientists like Isaac Newton and Christiaan Huygens helped develop the idea of mass. Newton eventually made these rules work for all bodies in the universe.

These equations connect to many parts of our world. In electrodynamics, they describe how charged particles move in electric fields. The Lorentz force is a key equation used there. Even in space, these rules change slightly. Relativity shows that space can be curved by gravity. Today, scientists even use similar equations in quantum mechanics. These equations help describe the tiny waves that make up our world.

Equations of motion are mathematical functions used to describe how a physical system behaves as it moves through time. These equations act as a map for movement, using variables to track an object's behavior. Most commonly, these variables include spatial coordinates, which tell us where an object is in space, and time. In more complex systems, scientists might also use momentum components to understand the motion. In classical mechanics, these functions are defined within a flat Euclidean space. However, in the study of relativity, these functions must account for curved spaces.

There are two primary ways to categorize these descriptions: kinematics and dynamics. Kinematics is a simpler approach that focuses only on variables derived from position and time. It describes how things move without worrying about why they move. Dynamics is a much broader field because it considers the causes of motion. It takes into account forces, energy, and the momentum of particles. When scientists solve differential equations to find the motion of a system, they are working within dynamics.

When an object experiences constant acceleration, physicists use a specific set of kinematic equations known as the SUVAT equations. The name comes from the five key variables involved in the process. These include displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). To solve a problem using these equations, one often starts with a differential equation, such as Newton's second law, which is F = ma. By applying the definitions of these physical quantities, you can set up an equation to solve for the unknown movement.

The history of these ideas spans thousands of years of human thought. In ancient times, astronomers used algorithms to predict eclipses and solar events, but they did not use formal equations of motion. During the thirteenth century, scholars at universities in Oxford and Paris began developing the foundations of physics. At Merton College in Oxford, thinkers like Thomas Bradwardine and Nicholas Oresme explored how velocity and force related to distance. Later, in 1545, the Spanish theologian Domingo de Soto correctly identified that certain types of motion were proportional to time. He even noted that acceleration would be negative when an object moves upward.

Galileo Galilei significantly advanced the field through geometric deduction and observation. He was the first to prove that the path of a projectile is a parabola. Galileo also studied the pendulum, observing that its period of oscillation is independent of its mass. He discovered that the period varies with the square root of the pendulum's length. While Galileo understood momentum as the product of velocity and weight, the concept of mass was developed later by Huygens and Newton. Newton eventually expanded these ideas to apply to all bodies in the universe, moving beyond just Earth's gravity.

In modern science, these equations have evolved to describe many different phenomena. In the field of electrodynamics, the Lorentz force equation describes how charged particles move through electric and magnetic fields.

As our understanding of the universe grew, special and general relativity required modifications to these classical equations. These updates account for the finite speed of light and the way gravity curves spacetime. Even in the tiny world of quantum mechanics, equations of motion exist. In that field, they are differential equations of the wavefunction, which describes how a quantum state behaves over space and time.

Mathematically, an equation of motion is often expressed as a second-order ordinary differential equation. It is a function of the object's position, its velocity, and its acceleration. Velocity is the first time derivative of position, while acceleration is the second time derivative. For a particle moving in a straight line with constant acceleration, the math becomes much simpler. In these cases, the velocity increases linearly over time. This means the average velocity multiplied by the time interval can help determine the total distance traveled.