Studying mechanical motion, physics uses various quantities to describe its quantitative characteristics. It is also necessary for the practical application of the results obtained. In the article, we will consider what acceleration is and what formulas should be used to calculate it.
Determining the value through speed
Let's begin to reveal the question of what acceleration is, by writing a mathematical expression that follows from the definition of this value. The expression looks like this:
a¯=dv¯ / dt
In accordance with the equality, this is a characteristic that numerically determines how quickly the speed of a body changes in time. Since the latter is a vector quantity, the acceleration characterizes its complete change (modulus and direction).
Let's take a closer look. If the speed is directed tangentially to the trajectory at the point under study, then the acceleration vector shows in the direction of its change over the selected time interval.
It is convenient to use the written equality if the function is knownv(t). Then it suffices to find its derivative with respect to time. Then you can use it to get the function a(t).
Acceleration and Newton's law
Now let's look at what acceleration and force are and how they are related. For detailed information, you should write down Newton's second law in the usual form for everyone:
F¯=ma¯
This expression means that the acceleration a¯ appears only when a body of mass m moves, when it is affected by a non-zero force F¯. Let's consider further. Since m, which in this case is a 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.
Understanding the written formula in practice is easy. If a force of 1 N acts on a body with a mass of 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 equal to 1 m/s2.
The formula given in this paragraph is fundamental for solving various kinds of problems on the mechanical movement of bodies in space, including the movement of rotation. In the latter case, an analogue of Newton's second law is used, which is called the "moment equation".
The law of universal gravitation
We found out above that the acceleration of bodies appears due to the action of external forces. One of them is the gravitational interaction. It works absolutely between anyreal objects, however, it manifests itself only on a cosmic scale, when the masses of bodies are huge (planets, stars, galaxies).
In the 17th century, Isaac Newton, analyzing a huge number of results of experimental observations of cosmic bodies, came to the following mathematical expression for the expression for the interaction force F between bodies with masses m1and m 2 that are r apart:
F=Gm1 m2 / r2
Where G is the gravitational constant.
Force F in relation to our Earth is called the force of gravity. The formula for it can be obtained by calculating the following value:
g=GM / R2
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/s2. In accordance with the dimension, we have received a value called free fall acceleration. We study the issue further.
Knowing what the acceleration of fall g is, we can write the formula for gravity:
F=mg
This expression exactly repeats Newton's second law, but instead of an indefinite acceleration a, the constant value g for our planet is used here.
When a body is at rest on a surface, it exerts a force on that surface. This pressure is called body weight. To clarify, it is the weight, and not the mass of the body, that we measure whenwe get on the scales. The formula for its determination unambiguously follows from Newton's third law and is written as:
P=mg
Rotation and acceleration
Rotation of systems of rigid bodies is described by other kinematic quantities than translational movement. 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 acceleration characterizes a change, only not of speed, but of a similar angular characteristic ω. The value of ω is measured in radians per second (rad/s), so α is calculated in rad/s2.
If linear acceleration occurs due to the action of a force, then angular acceleration occurs due to its momentum. This fact is reflected in the moment equation:
M=Iα
Where M and I are the moment of force and the moment of inertia, respectively.
Task
Having become acquainted with the question of what acceleration is, we will solve the problem of consolidating the considered material.
It is known that a car has increased its speed from 20 to 80 km/h in 20 seconds. What was his acceleration?
First we convert km/h to m/s, we get:
20 km/h=201,000 / 3,600=5.556 m/s
80 km/h=801,000 / 3,600=22.222 m/s
In this case, instead of the differential, the speed difference should be substituted into the formula for determining the acceleration, that is:
a=(v2-v1) / t
Substituting both speeds and the known acceleration time into equality, we get the answer: a ≈ 0.83 m/s2. This acceleration is called the average.
