What is the relationship between the object mass and acceleration if the applied force and constant?

Updated March 14, 2018

By John Papiewski

In the late 1600s, Sir Isaac Newton published "Principia Mathematica," a book that connected the worlds of math and physics. Among other important ideas, he described the second law of motion – that force is equal to mass times acceleration or f = ma. Although it looks simple at first glance, the law has several important implications, including how objects move on Earth and in space. Fundamental laws such as this have allowed scientists to investigate nature accurately and engineers to build machines that work.

Force equals mass times acceleration or f = ma.

Force is a physical quantity you deal with in everyday life. It takes force to open a door, lift a child, or crack an egg. It is a pull or push exerted by one object on another; the objects can be anything from protons and electrons all the way up to planets and galaxies. The pull or push may come from direct contact or, in the case of gravity, electricity and magnetism, from a distance. Scientists measure force in units called newtons, where one newton is the force needed to accelerate a 1-kilogram mass one meter per second squared.

When a hockey puck slides across the ice, it does so at a fairly constant speed until it hits the goal or a player’s stick. Although it’s moving, it’s not accelerating. Acceleration comes only from a change in speed. When an object gains speed, its acceleration is positive; when speed is lost, acceleration is negative. You measure speed in units of distance divided by time, such as miles per hour or meters per second. Acceleration is the change in speed divided by the time the speed takes to change, so it is meters per second per second, or meters per second squared.

The mass of an object is a measure of how much matter it contains. A rubber ball has less mass than a lead ball of the same size because it has less matter in it, fewer atoms and fewer of the protons, neutrons and electrons that make up the atoms. Mass also resists the effort to push or pull it; a ping-pong ball is easy to pick up and toss; a garbage truck is not. The truck is more massive than the ping-pong ball by many thousands of times. The standard unit for mass is the kilogram, about 2.2 pounds.

Mass is a simple kind of quantity. You can have large masses, tiny masses and in-between masses. That’s about it. Scientists call simple quantities scalars because one number will describe it. Force and acceleration, however, are more complicated. They have both a size and a direction. A TV weather forecaster, for example, talks about a wind coming from the west at 20 miles per hour. This is the velocity (speed) vector of the wind. To fully describe a force or acceleration, you need both the amount and the direction. For example, on a snowy day, you pull a child’s sled in the forward direction with a force of 50 newtons, and it accelerates in the same direction at 0.5 meters per second squared.

Newton’s second law of motion seems simple enough: Push on an object of a certain mass, and it accelerates based on the amount of force and mass. A small force with a large mass results in a slow acceleration, and a large force with a small mass gives a fast acceleration. What happens when there’s no force? A force of zero on any mass gives zero acceleration. If the object is standing still, it remains still; if it’s moving, it continues to move at the same speed and direction. Keep in mind that several forces can be involved at the same time. For example, you tie a rope around a boulder and pull with all your might. There are force and mass, but the boulder doesn’t budge, so acceleration is zero. The force of friction between the boulder and the ground cancels out the force of your pull. You need a much bigger force, such as from a tractor, to move the boulder.

Isaac Newton's First Law of Motion states, "A body at rest will remain at rest, and a body in motion will remain in motion unless it is acted upon by an external force." What, then, happens to a body when an external force is applied to it? That situation is described by Newton's Second Law of Motion.

According to NASA, this law states, "Force is equal to the change in momentum per change in time. For a constant mass, force equals mass times acceleration." This is written in mathematical form as F = ma

F is force, m is mass and a is acceleration. The math behind this is quite simple. If you double the force, you double the acceleration, but if you double the mass, you cut the acceleration in half.

Newton published his laws of motion in 1687, in his seminal work "Philosophiæ Naturalis Principia Mathematica" (Mathematical Principles of Natural Philosophy) in which he formalized the description of how massive bodies move under the influence of external forces.

Newton expanded upon the earlier work of Galileo Galilei, who developed the first accurate laws of motion for masses, according to Greg Bothun, a physics professor at the University of Oregon. Galileo's experiments showed that all bodies accelerate at the same rate regardless of size or mass. Newton also critiqued and expanded on the work of Rene Descartes, who also published a set of laws of nature in 1644, two years after Newton was born. Descartes' laws are very similar to Newton's first law of motion.

Newton's second law says that when a constant force acts on a massive body, it causes it to accelerate, i.e., to change its velocity, at a constant rate. In the simplest case, a force applied to an object at rest causes it to accelerate in the direction of the force. However, if the object is already in motion, or if this situation is viewed from a moving inertial reference frame, that body might appear to speed up, slow down, or change direction depending on the direction of the force and the directions that the object and reference frame are moving relative to each other.

The bold letters F and a in the equation indicate that force and acceleration are vector quantities, which means they have both magnitude and direction. The force can be a single force or it can be the combination of more than one force. In this case, we would write the equation as ∑F = ma

The large Σ (the Greek letter sigma) represents the vector sum of all the forces, or the net force, acting on a body.

It is rather difficult to imagine applying a constant force to a body for an indefinite length of time. In most cases, forces can only be applied for a limited time, producing what is called impulse. For a massive body moving in an inertial reference frame without any other forces such as friction acting on it, a certain impulse will cause a certain change in its velocity. The body might speed up, slow down or change direction, after which, the body will continue moving at a new constant velocity (unless, of course, the impulse causes the body to stop).

There is one situation, however, in which we do encounter a constant force — the force due to gravitational acceleration, which causes massive bodies to exert a downward force on the Earth. In this case, the constant acceleration due to gravity is written as g, and Newton's Second Law becomes F = mg. Notice that in this case, F and g are not conventionally written as vectors, because they are always pointing in the same direction, down.

The product of mass times gravitational acceleration, mg, is known as weight, which is just another kind of force. Without gravity, a massive body has no weight, and without a massive body, gravity cannot produce a force. In order to overcome gravity and lift a massive body, you must produce an upward force ma that is greater than the downward gravitational force mg.

Newton's second law in action

Rockets traveling through space encompass all three of Newton's laws of motion.

If the rocket needs to slow down, speed up, or change direction, a force is used to give it a push, typically coming from the engine. The amount of the force and the location where it is providing the push can change either or both the speed (the magnitude part of acceleration) and direction.

Now that we know how a massive body in an inertial reference frame behaves when it subjected to an outside force, such as how the engines creating the push maneuver the rocket, what happens to the body that is exerting that force? That situation is described by Newton’s Third Law of Motion.

Additional reporting by Rachel Ross, Live Science contributor.

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