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Calculating Acceleration with Force and Mass

By Kathleen Knowles, 23 Sep 2020

As is usually the case in mathematics and physics, formulas and experiments usually begin with curiosity --- even if it's something profoundly simple. What caused the apple to fall from the tree? Why did the ball move faster when it rolled downhill? These simple questions evolve into more complex questions. How do we measure the acceleration of the ball and apple? Someone already answered this question for us.

We use force and mass.

Mass

Every object in this universe has mass, except photons (the particle that carries light). Mass is the total amount of matter in an object. In an atom, protons, neutrons, and electrons account for its total mass.

A common misconception is that mass equals weight. This is not true. Mass is more fundamental, a total count of an object's matter expressed in the form of kilograms. Weight is a derivative of mass. You can calculate weight by multiplying the mass by gravity. An object's weight is different on other planetary bodies where the gravity varies. But, the object will always retain the same mass.

Force

Force is the push and/or pull acting on an object. A force acting on an object makes the object move, accelerate, stop, slow, or change direction. Force has magnitude and direction. We measure a force's magnitude in Newtons or Kgm/s2 (Kilogram meter per second squared), named after the father of physics, Isaac Newton. A force's direction is measured in degree or radian.

Acceleration

Acceleration is an object's change in velocity. A ball rolling downhill experiences an acceleration because it goes faster and faster as the ball rolls. A ball rolling on a level field is a deceleration because the ball gets slower and slower as the ground applies friction to the ball, which is also amplified by gravity. We measure acceleration in meter per second squared (m/s2)

Calculating Acceleration

Newton's Second Law

Mathematician and physicist Sir Isaac Newton the groundwork for the basic principles of physics. He developed three laws of motion in his book, "Principia Mathematica Philosophiae Naturalis."

His second law, in particular, is what makes our calculation work. In its pure form, the law states, "The alteration of motion is ever proportional to the motive force impress'd, and is made in the direction of the right line in which that force is impress'd."

What this means is that the vector sum of the forces (F = newtons) applied on an object is equal to its mass (m = kilograms) times its acceleration (a = meter/second).

In the real world, many different forces are acting on an object, even if this force is standing still. Wind direction and strength, gravity, and ground friction are some of the many factors. The vector sum means the total of the force's direction after it's all added up.

In an academic sense, we often gloss over this variable as a constant. Determining the source and magnitude of forces acting on an object can be much more challenging than executing the equation.

The Equation

The second law of Newton leads us to a beautifully simple equation:

F = m x a

Now, because we're solving for acceleration (a), we can rearrange that equation to:

a = F / m

Yes! It's that simple.

Example

Rolling Ball

We're going to use the question we asked earlier as an example.

A ball with a mass of 10 kg is barreling downhill with the total force of 50 N. Assuming the direction of the total force is perpendicular to the hill's slope, what's the ball's acceleration?

From that question, we can conclude that:

F = 50N

m = 10kg

Now, we're going to input that data into the equation.

a = F / m

a = 50 / 10

a = 5 m/s2

There we have it! The acceleration of the ball is 5 m/s2.

Fighter Jet

An F-35 fighter jet is summoned from the USS Eisenhower to inspect a certain event in the Atlantic Ocean. In mid-flight, the fighter jet has a massive thrust power of 200.000 N propelling the jet forward. The F-35 mass is 16.000 kg. Assuming the vector sum of forces from aerodynamic drag, gravity, and the fighter jet's thrust is indeed 200.000 N, calculate the acceleration of the fighter jet.

From that question, we can conclude that:

F = 200.000 N

m = 16.000 kg

Now, we're going to input that data to the equation:

a = F / m

a = 200.000 / 16.000

a = 12.5 m/s2

Just as easy as the first example! The acceleration of the fighter jet is 12.5 m/s2.

Conclusion

Mass is the total amount of matter in an object, measured in kilograms (kg). Force is the push or pull on an object, measured in newton (N). Acceleration is the rate of change in velocity experienced by an object, measured in meter per second squared (m/s2). The second law of Newton pioneered the equation to calculate acceleration. However, because determining the vector sum in real life is too complex, teachers make the vector sum in academic problems a constant, so students have an easier time solving it. The equation is F = m x a -- ​don't forget it!

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