Given
Mass and force Mass
lb
Net force magnitude
pdl
Results
Acceleration magnitude
ft/s²

# Magnitude of Acceleration Calculator

By Wojciech Sas, PhD candidate

Welcome to our magnitude of acceleration calculator - a fantastic tool that knows how to find the magnitude of acceleration. But that's not the end of the story. Read the article 📰 to learn:

• What does acceleration mean?
• Is acceleration a vector ↗?
• Is there a single magnitude of acceleration formula?
• How to find the acceleration from the velocity difference?

If you've asked yourself one of these questions recently 🤔, this is the best place to find the answer!

## What does acceleration mean?

According to the definition - acceleration is a rate of velocity change. The SI unit of acceleration is `m/s²`. In Physics, the velocity is a vector ↗, so a natural question arises - Is acceleration a vector ↗ as well?

Absolutely! How to check it? Let's say we have a car 🚗 that accelerates at `m/s²`. Is it enough information? - Not yet. To visualize, let's draw a coordinate system and an accelerating car. To do so, we also need to know the vehicle's direction and starting point because the acceleration IS a vector. If we say that it accelerates north-west, from point `(0,3)` at `2 m/s²`, the description is much better.

In our case, the value `2 m/s²` is the magnitude of acceleration. What is a vector magnitude in physics? For simplicity, we can say it's just a number. In general, we should specify all of the vector's properties, but it's usually just enough to give the magnitude.

Imagine a free-falling object 🍏. When we say that it accelerates at a rate of `9.81 m/s²`, it's just enough to visualize as a gravitational force always attracts to the object's center (which always means downwards from our perspective 🌎).

## How to find the magnitude of acceleration?

There are a few ways to estimate the magnitude of acceleration. We implement three of them in this magnitude of acceleration calculator:

1. According to Newton's second law, the acceleration `a` is proportional to the net force `F` and inversely proportional to the object's mass `m`. So how to calculate the magnitude of acceleration? By analogy - the magnitude of the acceleration is proportional to the magnitude of this force. Let's convert these words to the magnitude of acceleration formula:

`|a| = |F| / m`.

Here, bolded symbols represent vectors, and vertical lines denote a vector's magnitude, the absolute value, which is always positive (or equals zero).

2. As acceleration is a vector, we can always get this quantity by summing up the acceleration's components. We can use a simple addition of vectors. In general, for two components, we can write:

`a = a1 + a2`.

In the Cartesian plane, we can use x- and y-coordinates (`ax, ay`), and if we are working with a circular motion, it's worth implementing tangential and centripetal components (`at, ac`).

In either case, you receive a vector. And how to find the magnitude of acceleration? As both components are perpendicular to each other, square the values and work out the square root of the sum:

`|a| = √(|a1|2 + |a2|2)`.

In three-dimensional space the magnitude of acceleration formula extends to:

`|a| = √(|a1|2 + |a2|2 + |a3|2)`.
3. From the definition, acceleration is a rate of velocity change. If the initial velocity is `v0` and the final velocity is `v1`, the acceleration arises as to the difference of these vectors divided by time interval `Δt`:

`a = (v1 - v0) / Δt`.

Are you wondering how to calculate the magnitude of acceleration in this case? Don't worry, we explain in the next section in detail.

## How to find the acceleration from the velocity difference?

First things first - both acceleration and velocity are vectors. From the previous section, we know that the acceleration results from subtracting the final and the initial velocity divided by the time difference.

Imagine a sphere 🏐 in the Cartesian coordinates system. The initial velocity is `v0 = [-3,4] m/s`, and the final velocity equals `v1 = [3,2] m/s`. The velocity changed in time interval `Δt = 5 s`. We can ask two questions: What is the acceleration? and How to calculate the magnitude of acceleration? Let's find out:

• Evaluate the velocities' difference. For vectors, subtract each of the coordinates separately:

`v1 - v0 = [3,2] - [-3,4] = [3-(-3), 2-4] = [3+3, 2-4] = [6, -2] m/s`;
• Divide both components by time difference: `[3/5, -2/5] = [1.2, -0.4]`;

• The result is our acceleration: `a = [1.2, -0.4] m/s²`.

So how to find the magnitude of the acceleration? Let's use the formula with acceleration coordinates:

• Square each of the components: `(1.2)² = 1.44`, `(-0.4)² = 0.16`;

• Add these numbers: `1.44 + 0.16 = 1.6`;

• Estimate the square root of this value: `√(1.6) = 1.265`. We will stick with four significant figures; and

• That's all! The magnitude of the acceleration is `1.265 m/s²`.

## How to use the magnitude of the acceleration calculator?

Depending on your input data, there are three ways to calculate the magnitude of acceleration. Check which variables you have in your exercise at the beginning, and follow these steps:

1. Choose the appropriate option (depending on the input data);

2. Enter the values:

• Mass and force;
• The components of the acceleration (2-D or 3-D); or
• Time difference, and both the initial and final velocities' coordinates.
3. The end! The result appears immediately at the bottom 🎉.

Additionally, you can check the `Advanced mode` for the "how to find the acceleration from the velocity difference" option. You can check there the magnitudes of both velocities, as well as the components of the acceleration.

Wojciech Sas, PhD candidate