# How to calculate acceleration? (formulas and examples)

In physics, acceleration It is the name given to a vector magnitude (it has direction) that indicates the variation in speed of a moving body as time passes. It is normally represented by the letter a and its unit of measurement, in the International System it is meters per second squared (m / s2).

The origin of this concept comes from Newtonian mechanics, whose postulates ensure that an object will always move in a rectilinear way unless forces that lead to acceleration impinge on it, resulting in a curved path.

This means that an object in rectilinear motion can only vary its speed if a force acts on it that causes an acceleration: in the same direction of its trajectory (acceleration: it gains speed) or in the opposite direction of it (deceleration: it loses speed) .

### Formula for calculating acceleration

Thus, classical mechanics defines acceleration as the variation of speed over time and proposes the following formula:

a = dV / dt

Where to will be acceleration, dV the difference in speeds and dt the time at which the acceleration occurs. Both variables are defined as follows:

dV = VF – Vi

Where VF will be the final speed and Vi the initial velocity of the mobile. It is important to observe this order to reflect the direction of the acceleration: there can be a positive acceleration (gain speed) or negative (lose speed). Plus:

dt = tf – ti

Where tF will be the end time and ti the initial time of the movement. Unless otherwise stated, the start time will always be 0 seconds.

### Acceleration relative to force

On the other hand, Newtonian mechanics establishes for a body of constant mass (m), contemplated by an inertial observer, a relationship of proportionality with respect to the force applied to the object (F) and the acceleration obtained (a), that is:

F = m. to

This relationship is valid in any inertial reference system and allows the acceleration to be calculated with the following formula:

a = F / m

This formulation obeys Newton’s second Law (Fundamental Law of Dynamics).

### Acceleration calculation examples

1. A racing car increases its speed at a constant rate of 18.5 m / s to 46.1 m / s in 2.47 seconds. What will its average acceleration be?

a = dv / dt = (vF – vi) / (tF – ti) Where vF = 46.1 m / s, vi = 18.5 m / s, tF = 2.47 s, ti = 0 s.

So: a = (46.1 – 18.5) / 2.47 = 11.17 m / s2

1. A motorcyclist travels at 22.4 m / s and realizes that he has made the wrong route. Hit the brakes and the motorcycle comes to a stop in 2.55 seconds. What will its slowdown be?

a = dv / dt = (vF – vi) / (tF – ti), where VF = 0 m / s, vi = 22.4 m / s, tF = 2.55 s, ti = 0 s.

So: a = (0 – 22.4) / 2.55 = -8.78 m / s2

1. A force of magnitude 10 newtons acts uniformly on a 2 kilogram mass. What will be the acceleration of the pushed object?

a = F / m, where F = 10 N, m = 2Kg.

Therefore:

a = 10/2 = 5 m / s2

1. Someone pulls a 400-kg piece of furniture to one side with a net force of 150 newtons. Another person is pushing it in the same direction with a force of 200 newtons, but there is a wind blowing in the opposite direction with a force of 10 newtons. What will be the acceleration obtained by the furniture?

We know that a = F / m, where the net force will be the sum of those in the same direction minus the one that opposes them: F = 150 N (person 1) + 200 N (person 2) – 10N (wind), which results in 340 N. We also know that m = 400 kg.

Later: a = 340 N / 400 kg = 0.85 m / s2

1. A remote-controlled airplane, with a mass of 10 kg, flies with an acceleration of 2 m / s2 heading north. Just then a wind is blowing to the east, with a force of 100 N. What will be the new acceleration of the plane that maintains its course?

Since the force of the wind is perpendicular to the direction of the plane’s motion, it will have no effect on its motion. It will continue to accelerate northward at 2 m / s2.

1. Two children, one weak and one strong, play tug of war, each at one end of the rope. The first pulls to the left with a force of 5 N and the second pulls in the opposite direction with a force of 7 N. Taking into account that 1 newton (N) is equal to 1 kilogram-meter / second squared ( kg-m / s2), what will be the acceleration reached by the body of the weaker child, when being pulled in the opposite direction by the other?

From F = ma we know that a = F / m, but we must find the net force, which will be 2 N (7 N for the strong child – 5 N for the weak child).

Then, we must find the mass, which for calculation purposes must be detached from the force that the weak child opposes, namely: 1 N = 1kg.m / s2, that is, it is the amount of force to mobilize one kilogram of mass at one meter per second squared.

Therefore, from 5N = 5kg.m / s2. Hence, m = 5Kg.

And finally, we know that a = 2N (F) / 5kg (m) = 0.4 m / s2

1. A fire truck increases its speed from 0 to 21m / s towards the East, in 3.5 seconds. What is its acceleration?

We know that: Vi = 0 m / s, VF= 21 m / s, t = 3.5 seconds. Therefore we apply the formula:

a = dv / dt = (vF – vi) / (tF – ti), that is, a = 21m / s / 3.5 s = 6 m / s2

1. A car slows down from 21m / s East to 7m / s East in 3.5.0 seconds. What is its acceleration?

Knowing that Vi = 21 m / s, VF= 7 m / s, t = 3.5 seconds, and that a = dv / dt = (vF – vi) / (tF – ti), it is simple to calculate it:

a = 7m / s – 21m / s / 3.5s = -4m / s2 that is, a negative acceleration (deceleration).