What is Newton's Second Law of Motion?

What is Newton's Second Law of Motion and why is it so important?
John Loeffler

When Isaac Newton developed his three Laws of Motion in 1666 (published in 1686), he gave the study of motion a theoretical framework that would serve as the basis of modern physics to this day. 

And though his work has been somewhat eclipsed by Albert Einstein, both in the field of gravity and in the popular imagination, his work is still critical to even the most trivial engineering projects, as well as the most audacious.

Newton's Laws of Motion are famous for their definition of inertia, and the well-known rule that every action will have an equal and opposite reaction. But more than anything, it's the math behind it all that makes Newton's work so revolutionary – and the law everyone tends to skip over is all about the math of motion.

Newton's Laws of Motion

Issac Newton's Laws of Motion can be defined as:

  1. An object at rest stays at rest, and an object in motion will stay in motion, at a constant speed in a straight line, unless another force acts upon it.
  2. The acceleration of an object depends on the mass of the object and the amount of force applied.
  3. Whenever one object exerts force on another object, the second object exerts an equal and opposite force on the first. 

Briefly, the first law states that when an object is at rest, you need to overcome its inertia in order to accelerate it. Likewise, when you want to stop an object in motion or steer it in another direction, you also need to overcome that object's inertia to slow it down to a resting state.

This is pretty intuitive for most people. If you are in a speeding car and you suddenly try to make a turn without decelerating, your car is probably going to shift over to the next lane, and you're going to miss your turn completely because you failed to overcome the car's inertia.

Likewise, it gets easier to make that turn when the car's speed is much less because the car's inertia is lower - somewhere between the inertia of the speeding car and one that is parked.

Finally, you cannot turn a parked car at all, not without applying force from the engine (or a push) to overcome the parked car's inertia, which keeps it in place.

Newton's Third Law, briefly, means that if an airplane is taking off, the weight of the plane pushes the wings of the plane down because of gravity. The air rushing beneath the wings, on the other hand, pushes up against the bottom of the wing, which generates lift.

There are countless other examples like this that you can find, but the one thing they will all have in common is that the math that governs the behavior and outcome of all of these interactions can be found in the Second Law of Motion.

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What is Newton's Second Law of Motion and Why Is It Important?

Newton's Second Law of Motion is that an object's acceleration depends on the mass of the object and the force applied. Sounds simple enough, but there's a lot more to it. 

First, we'll need to define a few terms for this to make sense. The first is velocity, which is a measure of how fast an object is moving at a given time. The second is mass, which is the physical measure of how much matter an object contains. When you multiply these two together, you get a third term, called an object's momentum.

Force, then, is defined to be equal to the change in an object's momentum divided by the change in time. This gives us two different equations that we can use to calculate a measure of force.

First, when dealing with an object of constant mass, you simply divide the change in velocity by the change in time, so:

Force = (v1 - v0) / (t1 - t0)

This is also the formula for acceleration, so we usually just say:

Force = mass x acceleration

But there are some very important circumstances where the mass of the object isn't constant, such as during a rocket launch or a commercial passenger jet, or pretty much whenever an object is burning through large amounts of fuel to generate velocity.

In these instances, we say:

Force = (m1v1 - m0v0) / (t1 - t0)

It's important to make that distinction since Newton's Second Law of Motion is especially important when we're doing things like launching rockets, or calculating how much fuel a plane will need to fly from one city to the next, all of which are important questions in engineering.

You might also have noticed that the two equations above are differential equations, which are a foundational component of calculus, which Newton himself (along with Gottfried Leibniz) developed and codified. In terms of calculus, we would write

Force = dm / dt

Where force is equal to the derivative of m, defined as the object's momentum, in relation to time, t.

How do we use Newton's Second Law in Engineering?

Have you ever watched a SpaceX rocket launch? Besides being one of the most incredible spectacles of our time, Newton's Second Law of Motion is essential to understanding how we can move a rocket as massive as the Falcon 9 against the pull of Earth's gravity to get it into orbit.

There are many other practical circumstances where we need to use Newton's Second Law of Motion to determine how objects are going to behave when a certain amount of force is applied.

Probably the most direct application of Newton's Second Law of Motion is in ballistics, which makes it possible to actually calculate the trajectory of a projectile with a high degree of accuracy.

The cannon had been in use for centuries before Newton was even born, perhaps the most famous early use of cannon was when the Ottomans used them to blow open the walls of Constantinople in 1453. But without Newton's Second Law of Motion, artillery officers pretty much pointed the cannon in the general direction of the target and performed ad hoc estimates, based on where projectiles landed, to narrow down their aim.

Newton's Second Law of Motion made more precise calculations of trajectories possible, making artillery far more lethal in the following centuries as officers could calculate where a cannonball or shell would land before it was even fired.

Less aggressive applications of Newton's Second Law of Motion would be important during the Industrial Revolution. With the introduction of the steam engine (and with it, locomotives, steamboats, and industrial factories) how to power an engine and how to use the force an engine produced to turn systems of gears through acceleration became just as important as the development of modern accounting practices to a factory owner.

While the factory owners may not have known how to do all that math, they had engineers who did, because they had Newton's Second Law of Motion and the math that it provided them. It's perhaps not a coincidence that the Industrial Revolution in England started less than a century after Newton published the Philosophiæ Naturalis Principia Mathematica.

In a preindustrial world, it took time to circulate this material outside of the academy, but circulate it did. Those who studied and learned the classical mechanics that the Second Law of Motion inspired wasted no time using it to transform the world through machinery.

More than anything, that is what makes Newton's Second Law of Motion arguably more important than the other two, since the second law is the one that showed us how to calculate what it would take to move mountains.