What is the relationship between the mass of an object and the amount of force needed to push it?

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.

See also:

• Newton's Laws of Motion
• Inertia & Newton's First Law of Motion

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Updated April 24, 2017

By Meg Kramer

Sir Isaac Newton first discovered the physical principles underlying the relationship between mass and matter in the late 1600s. Today, mass is considered to be a fundamental property of matter. It measures the amount of matter in an object, and also quantifies the object’s inertia. The kilogram is the standard unit of measurement for mass.

While mass is measured in kilograms, a unit that is also used for weight, there is a difference between mass and weight. An object’s weight (w) is defined by its mass (m) times the acceleration of gravity (g), expressed in the formula w = mg. This means that when gravity changes, so does an object’s weight. For example, even if your mass remains constant, your weight on Earth is six times greater than your weight would be on the moon, which has a weaker gravitational pull.

Galileo first postulated the concept of inertia in the 17th century, and in his first law of motion, Sir Isaac Newton further developed Galileo’s observations. According to the first law, without the intervention of an external force, objects in motion will continue to move at the same speed in a straight line. Objects at rest, on the other hand, will remain at rest unless an external force moves them. This tendency to resist changes in motion is known as “inertia,” and it is directly related to the object’s mass. The more massive an object is, the more it resists changes in its motion.

Momentum occurs when an object is in motion, and can be transferred from one object to another when the two collide. It is the combination of mass and velocity, and has a directional quality, pointing in the direction of the object’s motion. There is a direct relationship between mass and momentum, meaning that the greater an object’s mass, the greater its momentum will be. Increasing an object’s velocity will also result in increased momentum.

When an external force acts on an object, the change in the object’s motion will be directly related to its mass. This change in motion, known as acceleration, depends upon the object’s mass and the strength of the external force. The relationship between force (F), mass (m) and acceleration (a) is described in the equation F = ma. This equation means that a new force acting on a body will change velocity, and conversely, a change in velocity will generate a force.