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Magnitude of Acceleration Calculator

Created by Wojciech Sas, PhD
Reviewed by Steven Wooding
Last updated: Jan 18, 2024

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!

🔎 Head on to our acceleration calculator for even more examples!

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.

🔎 You can read more about evaluating a vector's magnitude in Omni's vector magnitude calculator.

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 🌎).

After this short introduction, it's high time we asked about how to calculate the magnitude of acceleration. Jump to the next sections to get the answer.

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 aa is proportional to the net force FF and inversely proportional to the object's mass mm. 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=Fm\qquad |\bold{a}| = \left|\frac{\bold{F}}{m}\right|

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

  1. 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\qquad \bold{a} = \bold{a_1} + \bold{a_2}

In the Cartesian plane, we can use x- and y-coordinates (ax\bold{a_x}, ay\bold{a_y}), and if we are working with a circular motion, it's worth implementing tangential and centripetal components (at\bold{a_t}, ac\bold{a_c}).

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=a12+a22\qquad |\bold{a}| = \sqrt{|\bold{a_1}|^2 + |\bold{a_2}|^2}

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

a=a12+a22+a32\qquad |\bold{a}| = \sqrt{|\bold{a_1}|^2 + |\bold{a_2}|^2 + |\bold{a_3}|^2}
  1. From the definition, acceleration is a rate of velocity change. If the initial velocity is v0v_0 and the final velocity is v1v_1, the acceleration arises as to the difference of these vectors divided by time interval Δt\Delta t:
a=v1v0Δt\qquad a = \frac{v_1-v_0}{\Delta t}

Are you wondering how to calculate the magnitude of acceleration in this case? Don't worry; we will 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

To understand how to find the rate of change, you can also check our rate of change calculator.

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.


How do I compute the magnitude of acceleration from velocity vectors?

To calculate the magnitude of the acceleration from the velocity vectors, follow these easy steps:

Given an initial vector vi = (vi,x, vi,y, vi,z) and a final vector vf = (vf,x, vf,y, vf,z):

  1. Compute the difference between the corresponding components of each velocity vector:

    vf − vi = (vi,x − vf,x, vi,y − vf,y, vi,z − vf,z)

  2. Divide each difference by the time needed for this change Δt to find the acceleration components ax, ay, az.

  3. Compute the square root of the sum of the components squared:

    |a| =√(ax² + ay² + az²)

What is the magnitude of the acceleration caused by a force of 50 N on a mass of 100 kg?

The magnitude of the acceleration would be 0.5 m/s2. To find this result:

  1. Be sure you are dealing with the magnitude of the force. In this case, we can proceed as we don't see any component.
  2. Divide the magnitude of the force by the mass: |F|/m = 50 N/100 kg = 0.5 N/kg.
  3. The result is the magnitude of the acceleration in meters per seconds square: |a| = |F|/m = 0.5 m/s2.

How do you calculate the magnitude of the acceleration?

You can compute the magnitude of the acceleration in different ways, depending on the setup of your problem.

  1. If you know the mass m and the force F, the magnitude of the acceleration would be |a| = |F|/m.
  2. If you know the components of the acceleration, it would be the square root of the sum of the squared components: |a| =√(ax² + ay² + az²).
  3. If you know the time and the initial and final velocity vector, compute the acceleration by dividing the component-wise difference of the vectors by the time, and then follow point 2.

What is the magnitude of the acceleration?

The magnitude of the acceleration is a measure of the absolute change in your speed or force you are subjected to, regardless of the direction of the movement. As acceleration is a vector — that is, it depends on a direction — you can compute the acceleration (the variation in speed) for each component or all the components regardless of the direction. In the latter case, you must apply the Pythagorean theorem to compute the proper magnitude of the total acceleration.

Wojciech Sas, PhD
Mass and force
The magnitude of acceleration formula using net force and mass.
Net force magnitude
Acceleration magnitude
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