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What Are Equations of Motion?

By Valerie Clark
Updated Feb 01, 2024
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Equations of motion are used to determine the velocity, displacement or acceleration of an object in constant motion. Most applications of the equations of motion are used to express how an object moves under the influence of a constant, linear force. Variations of the basic equation are used to account for objects moving on a circular path or in a pendulum configuration.

An equation of motion, also referred to as a differential equation of motion, mathematically and physically relates Newton’s second law of motion. The second law of motion, according to Newton, states that a mass under the influence of a force will accelerate in the same direction as the force. Force and magnitude are directly proportional, and force and mass are inversely proportional.

Standard equations of motion involve five variables. One variable is for the starting and ending position of the object, also known as displacement. Two variables represent the initial and final velocity measurements, respectively known as the change in velocity. The fourth variable describes acceleration. The fifth variable stands for the time interval.

The classic equation to solve the linear acceleration of an object is written as the change in velocity divided by the change in time. The law of motion equation typically is set up using three kinetic variables: velocity, displacement and acceleration. Acceleration can be solved for by using velocity and displacement as long as the second law of motion applies to the problem.

When an object is in constant acceleration along a rotational trajectory, the equations of motion are different. In this situation, the classic equation for circular acceleration of an object is written using the initial and angular velocities, angular displacement and angular acceleration.

A more complicated application of the equations of motion is the pendulum equation of motion. The basic equation is known as Mathieu’s equation. It is expressed using the gravity constant for acceleration, the length of the pendulum and the angular displacement.

There are several assumptions that must be satisfied to use such an equation for a problem involving a pendulum configuration. The first assumption is that the rod that connects the mass to the axis point is weightless and remains taut. The second assumption is the motion is limited to two directions, back and forth. The third assumption is that the energy lost to air resistance or friction is negligible. Variations of the basic equation are used to account for infinitesimal oscillations, compound pendulums and other configurations.

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Discussion Comments

By everetra — On Aug 15, 2011

@nony - The most fun I’ve ever had in physics was with problems having to do with projectile motions.

The reason is that I saw so many useful applications – like how to write a tank game, which I did at least on a very simple level, in BASIC.

Once you know velocity and angles you can compute the X displacement and the Y displacement of the projectile (a missile in this case) at various intervals. Also you have to take into account the effects of gravity; obviously the missile has to fall to the ground.

One of the benefits of understanding physics, if you are writing games, is that the games are much more realistic. They simulate motion in reality which makes the games more enjoyable to play, in my opinion.

By nony — On Aug 15, 2011

@David09 - Yes, I discovered that very thing when I had to help my daughter with her 12th grade physics glass.

As the parent I had to learn the material myself of course and that was also one of the first discoveries I made. It also explains why, in order to solve the physics motion equations, I had to sometimes use trigonometry to get the correct answer.

If you are factoring in direction, you need a way of determining final direction in your answer, and the trigonometry functions will help get you there.

By David09 — On Aug 14, 2011

One of the most confusing things to understand from equations of motion, at least for me at first, was the definition of velocity.

In physics equations, velocity is not speed alone, as I used to think. Velocity defines both speed and direction. For example, if you say a car is going 55 miles per hour, you’re not describing velocity, you’re describing speed.

If, however, you say that a car is going 55 miles per hour north west, then you’re describing velocity. You’ve added direction into the mix, which is part and parcel of velocity.

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