Studying mechanical motion, various quantities are used in physics to describe its quantitative characteristics. It is also necessary for applying the obtained results in practice. In the article, we will consider what acceleration is and by what formulas it should be calculated.
Definition of value through speed
We begin to uncover the question of what acceleration is, by writing a mathematical expression that follows from the definition of a given quantity. The expression looks like this:
a¯ = dv¯ / dt
In accordance with equality, this is a characteristic that numerically determines how quickly the speed of the body changes over time. Since the latter is a vector quantity, acceleration characterizes its complete change (modulus and direction).
Let's consider in more detail. If the speed is directed along the tangent to the trajectory at the studied point, then the acceleration vector shows in the direction of its change over the selected period of time.
It is convenient to use the written equality if the function v (t) is known. Then it is enough to find its time derivative. Then we can use it to obtain the function a (t).
Acceleration and Newton's Law
Now consider what acceleration and force are, and how they are related. To obtain detailed information, the second Newtonian law should be written in the form familiar to all:
F¯ = m * a¯
This expression means that the acceleration a¯ appears only when a body of mass m is moving, when a nonzero force F¯ affects it. Let's consider further. Since m, which in this case is the characteristic of inertia, is a scalar quantity, the force and acceleration are directed in the same direction. In fact, mass is only a coefficient that connects them.
To understand the written formula in practice is simple. If a force of 1 N exerts an effect on a body weighing 1 kg, then for every second after the start of movement, the body will increase its speed by 1 m / s, that is, its acceleration will be 1 m / s 2 .
The formula given in this paragraph is fundamental for solving various kinds of problems on the mechanical movement of bodies in space, including the motion of rotation. In the latter case, they use an analog of Newton’s second law, which is called the “equation of moments”.
Law of gravity
Above, we found out that the acceleration of bodies appears due to the action of external forces. One of them is gravitational interaction. It acts absolutely between any real objects, however, it manifests itself only on a cosmic scale, when the masses of bodies are huge (planets, stars, galaxies).
In the XVII century, Isaac Newton, analyzing a huge number of results of experimental observations of cosmic bodies, came to the following mathematical notation of the expression for the interaction force F between bodies with masses m 1 and m 2 that are at a distance r from each other:
F = G * m 1 * m 2 / r 2
Where G is the gravitational constant.
Force F in relation to our Earth is called gravity. The formula for it can be obtained if we calculate the following value:
g = G * M / R 2
Where M and R are the mass and radius of the planet, respectively. If we substitute these values, we get that g = 9.81 m / s 2 . In accordance with the dimension, we got a value called acceleration of gravity. We study the question further.
Knowing what the acceleration of falling g is, we can write the formula for gravity:
F = m * g
This expression exactly repeats Newton’s second law, only instead of the indefinite acceleration a, here we use the constant g for our planet.
When the body is at rest on a surface, it exerts a force on this surface. This pressure is called body weight. We will clarify that it is weight, not body weight, that we measure when we stand on the scales. The formula for its determination clearly follows from Newton’s third law and is written as:
P = m * g
Rotation and acceleration
The rotation of systems of solids is described by other kinematic quantities than translational displacement. One of them is angular acceleration. What does it mean in physics? The following expression will answer this question:
α = dω / dt
Like linear acceleration, angular characterizes a change, not just of speed, but of a similar angular characteristic ω. The value of ω is measured in radians per second (rad / s), therefore α is calculated in rad / s 2 .
If linear acceleration arises due to the action of a force, then angular acceleration arises due to its momentum. This fact is reflected in the equation of moments:
M = I * α
Where M and I are the moment of force and the moment of inertia, respectively.
Task
Having become acquainted with the question regarding what acceleration is, we will solve the problem of consolidating the considered material.
It is known that the car in 20 seconds increased its speed from 20 to 80 km / h. What was its acceleration equal to?
First, convert km / h to m / s, we get:
20 km / h = 20 * 1,000 / 3,600 = 5.556 m / s
80 km / h = 80 * 1,000 / 3,600 = 22,222 m / s
In this case, in the formula for determining acceleration, instead of the differential, you should substitute the speed difference, that is:
a = (v 2 -v 1 ) / t
Substituting both speeds and the known acceleration time into the equality, we get the answer: a ≈ 0.83 m / s 2 . This acceleration is called average.