## Concept and basic terms

Speed is understood as a quantity that determines the speed and direction of movement of a material point in the selected frame of reference. The term is widely used in mathematics, physics, chemistry. So, with its help, reactions, temperature changes, movement of bodies are described, used as a derivative of the value under consideration.

The word "speed" comes from the Latin "velocitas" meaning movement. As a unit of measurement, according to the International System of Units (SI), a meter divided by a second (m / s) is chosen for it. The speed is denoted by the letter V, regardless of the science in which it is used. The simplest formula used to determine the value is as follows: V = S: t. **Where: **

- S - distance (path) traversed by a material point or body (m);
- T - time during which she covered the path (s).

This is a generalized equation, but at the same time allows you to get an idea of the concept. This inequality is often referred to as the equation of the path. The formula is used for calculation only if the movement does not change over the entire area of interest.

**For the first time, the expression is introduced to students in mathematics lessons in the fifth grade **... The teacher offers to learn how to solve simple problems of finding a characteristic with a known length of the path traveled and the time spent on it. For example, a car covered 16 kilometers in four hours. It is necessary to find out how fast he was moving. The solution to the problem comes down to two steps. In the first, all specified values are converted into SI: 4 hours = 240 minutes = 10240 seconds; 16 kilometers = 16,000 meters. In the second step, the data is substituted into the formula and the answer is calculated: V = 16000/10240 = 1.6 m / s.

But, in addition to uniform motion, that is, in which the speed is constant, there are also other types of displacement. You cannot use the generalized equation for them. Each type of movement has its own formula. **The existing speed is divided into the following types: **

- uneven;
- medium;
- uniformly variable;
- translational;
- rotational;
- accelerated.

## Equally accelerated movement

If over time the position of the body changes relative to objects at rest, then it is considered that it is moving. In this case, speed is used as the main parameter describing the movement. The movement of a body or point can be represented as a line repeating the path of passage. It is called a trajectory. If the line is straight, then the movement is considered straight.

Uneven movement is characterized by movement along a different trajectory with an inconsistent speed value. In this case, the change in position can be uniformly accelerated, that is, the parameter at equal intervals increases or decreases by the same value. An example is the fall of a stone.

At an arbitrary point, the speed of movement is equal to the acceleration of gravity.

Thus, if the vectors V and accelerations A lie along a straight line, then in projections such a direction can be considered as algebraic quantities. With uniformly accelerated motion along a straight trajectory, the speed of a point is calculated by the formula: V = V0 + A * t. **Where: **

- V0 - initial speed;
- A - acceleration (has a constant value);
- t is the time of movement.

**This is the basic formula in physics. **... On the graph, it is depicted as a straight line v (t). The ordinate is time, and the abscissa is speed. Having built a graph, the slope of the straight line can be used to determine the acceleration of point A. For this, the formula for finding the sides of a triangle is used: A = (v-v0) / t.

If the interval Δt is selected on the time axis, then it can be assumed that the motion will be uniform and described by some parameter equal to the instantaneous value in the middle of the segment. This instantaneous value is vector. It is numerically equal to the limit that the speed is trying to reach for a period of time tending to zero. In physics, this state is described by the formula for instantaneous velocity: V = lim (Δ s / Δ t) = r ^{-one }(t). That is, from a mathematical point of view, this is the first derivative.

Based on this, it can be argued that the movement is Δs = v * Δt. Since the product of acceleration and time is determined by the difference V -V0, the following will be correct: S = V0 * t + A * t

^{2}/ 2 = (V^{2}- V^{2}0) / 2 * A.

From this formula, you can derive an expression for finding the final speed of a material point: V = (V ^{2}0 - 2 * A * s) ^{½}... If at the initial moment V0 = 0, then the formula can be simplified to the form: V = (2 * A * s) ^{½}.

## Mean

In kinematics, the averaged parameter is used to find the characteristic. They use it to study the movement of a material point or any physical body. To determine the average speed, two quantities are used: scalar and vector. The first is the track movement, and the second is the movement.

Ground speed is defined as the ratio of the distance traveled by the body to the time spent on its passage: V = Σs / Σt.

In fact, the average value is found as the arithmetic mean of all speeds if the point under consideration moved the same time intervals. Otherwise, the found value will be a weighted arithmetic mean.

Mathematically, the average speed formula is written as follows: V (t + Δ t) = Δ s / Δ t = (s (t + Δ t) - s (t)) / Δ t. Considering that Δs depends on the length of the path that the point has covered during the time Δt, the correct record will be: Δ s = s (t + Δt) - s (t). If the elapsed time tends to zero, you get a formula that coincides with the expression for finding the instantaneous speed.

The vector of a material point is found from the ratio of the position of the body to the time interval: V (t + Δt) = Δr / Δt = (r (t + Δt) - r (t)) / Δt, where r is the radius vector. When the body performs uniformly rectilinear movement, then the equality will be fair: {V} = V.

For example, for the first half of the 100-meter path, the ball rolled at one speed for twenty seconds, and the second, with the other, for one minute. The average speed needs to be calculated. According to the formulas, the interval of movement on the first section of the path will be equal to: t1 = s / 2 * V1, and on the second t2 = s / 2 * V2. The solution to the problem will be: Vav = s / (t1 + t2) = s / (s / 2 * v1 + s / 2 * v2) = 2 * V1 * V2 / (V1 + V2) = 100 / (20 +60) = 1.25 m / s.

## Angular velocity

**This type appears when the body rotates around the axis. **... The trajectory is a circular motion. The main parameter taken into account when finding it is the angle of rotation (f). All elementary angular movements are vectors. The usual rotation is equal to the angle of rotation of the body df in a short period of time dt in the opposite direction from the clockwise movement.

In mathematics, the formula for finding the angular parameter is written as w = df / dt. Angular velocity is an axial quantity located along the instantaneous axis and coinciding with the translational rotation of the right screw. Uniform rotation, that is, a movement that rotates through the same angle, is called uniform. The angular velocity modulus is determined by the formula: w = f / t, where f is the angle of rotation, t is the time during which the rotation took place. Considering that Δf = 2p, the formula can be rewritten to the form: w = 2p / T, that is, using the period.

There is a relationship between the angular velocity and the number of revolutions: w = 2 * p * v. This concept is used to solve problems when describing non-uniform rotation. There is also an expression connecting the linear velocity with the angular velocity: v = [w * R], where R is the component drawn perpendicular to the radius vector. The unit of measurement for the parameter is the radian divided by the second (rad / s).

For example, it is necessary to determine the angular speed of the variator at the moment when the suspended mass travels a distance of 10 meters. The radius of the shoulder is 40 centimeters. At the initial moment, the suspension is at rest, and then begins to descend with an acceleration A = 0.04 m / s2.

Considering that the linear speed of the variator coincides with the movement of the load in a straight line, we can write: V = (2 * a * S) ½. The answer should be: V = (4 * 0.04 * 10) ½ = 1.26 m / s. The angular velocity is found by the formula: w = v / R, since R = 40 cm = 0.4 m, then W = 1.26 / 0.4 = 3.15 rad / s.

## Addition law

For different frames of reference for the movement of material points, there is a law connecting them to each other. According to him, the speed of something relative to the system at rest is determined by the sum of the force of displacement of the velocities in the moving region and a faster frame of reference with respect to the stationary one.

To understand the essence of the law, it is best to consider a simple example. Let a carriage move along the railway at a speed of 80 km / h. A passenger moves in this car at a speed of 3 km / h. Taking a stationary railway track as a reference system, it can be argued that the passenger's speed relative to it is equal to the sum of the speed of the carriage and the person.

If the movement of the car and the passenger occurs in the same direction, then the values are simply added, V = 80 + 3 = 83 km / h, in the opposite direction, V = 80−3 = 77 km / h are subtracted. But this rule will be true only when the movement occurs along one line. Therefore, if a person moves at an angle in the carriage, this factor should also be taken into account, since in essence the sought parameter is a vector quantity. In fact, two speeds are calculated: approach and withdrawal.

**The event under consideration occurs during the time Δt **... During this interval, a person will cover the distance ΔS1, while the car will be able to travel the path ΔS2. Using the law, the passenger's movement will be determined by the formula: ΔS = ΔS1 + ΔS2. A person's own movement relative to the railway track will be equal to V = ΔS1 / Δ t. Expressing the value from the formula for finding ΔS, you can find the speed of the car relative to the railroad: V2 = ΔS2 / Δt.

## Using an online calculator

There are services on the Internet that allow you to find a parameter even for those who do not know the formula or are poorly guided by the topic. With their help, you can solve quite complex tasks that require meticulous calculation and a considerable investment of time. Online calculation usually takes no more than a few seconds, and you don't have to worry about the reliability of the result.

Any user with an Internet connection and an installed web browser with support for Flash technology can use calculator sites. The services offering this kind of services do not require any registration or the provision of personal data. The system will automatically calculate the answer.

**Of the many sites, three are the most popular among consumers: **

- Help portal "Calculator".
- Allcalc.
- Fxyz.

All of them have an intuitive interface and, what is remarkable, on their pages they contain tables of all formulas used to solve problems, correct conventions and descriptions of calculation processes.

Calculating the speed of any body is straightforward. The main thing is to know the formulas and correctly determine the type of movement. In this case, you can always use the services of online calculators. Through them, solve the problem or check your calculations.

The concept of speed is widely used in science: mathematics, physics, mechanics. Schoolchildren begin to get to know him already in the third grade. This happens in more detail in grades 7-8. In a generally accepted sense, speed is a quantity that characterizes how quickly an object moves through space per unit of time. Depending on the application, the speed is indicated by different symbols.

1

How speed is denoted in mathematicsIn mathematics textbooks, it is customary to use the lowercase Latin letter v. Speed is related to the distance traveled and the time for which it has been traveled.

With uniform motion, the value v = S / t, where:

- S - the path traveled by the body,
- t is the time of movement.

2

How speed is denoted in physicsThe branch of physics called mechanics also studies speed. The designation of the velocity depends on whether it is a vector value or an ordinary one. In the first case, an arrow pointing to the right → is placed above the letter v. If there is no need to take into account the direction, then the usual symbol v is used.

3

Speed unitsIn the international system of units of measurement, it is customary to operate in meters per second (m / s). At the same time, the generally accepted units of measurement are kilometer per hour (km / h), knot (nautical mile per hour).

4

How the speed of light and sound is indicatedScientists have proven that the speed of light is the absolute value with which information and energy can move. This indicator is constant and equal to 299 792 458 ± 1.2 m / s. The Latin letter c was chosen as its symbol.

The speed of sound depends on the density and elasticity of the medium in which the sound waves propagate. It is measured in Mach. For example, supersonic speed ranges from Mach 1.2 to Mach 5. And everything above is called hypersonic speed.

Obviously, the symbol that denotes speed depends on the mathematical or physical meaning that this concept is filled with.