This article talks about how to find the average speed. A definition of this concept is given, and two important special cases of finding the average speed are also considered. A detailed analysis of problems on finding the average speed of a body from a tutor in mathematics and physics is presented.
Determination of average speed
Medium speed movement of a body is called the ratio of the distance traveled by the body to the time during which the body moved:
Let's learn how to find it using the following problem as an example:
Please note that in in this case this value did not coincide with the arithmetic mean of the speeds and , which is equal to:
m/s.
Special cases of finding the average speed
1. Two identical sections of the path. Let the body move with speed for the first half of the path, and with speed for the second half of the path. You need to find the average speed of the body.
2. Two identical intervals of movement. Let a body move with speed for a certain period of time, and then begin to move with speed for the same period of time. You need to find the average speed of the body.
Here we got the only case when the average speed coincided with the arithmetic mean of speeds on two sections of the route.
Let us finally solve the problem from All-Russian Olympiad schoolchildren in physics last year, which is related to the topic of our lesson today.
| The body moved with, and the average speed of movement was 4 m/s. It is known that during the last period of movement the average speed of the same body was 10 m/s. Determine the average speed of the body during the first s of movement. |
The distance traveled by the body is: 
m. You can also find the path that the body has covered in the last since its movement: m. Then, in the first since its movement, the body has covered a distance in m. Consequently, the average speed on this section of the path was: 
m/s.
Problems to find the average speed of movement are very popular at the Unified State Examination and the Unified State Examination in physics, entrance exams, and Olympiads. Every student must learn to solve these problems if he plans to continue his studies at a university. A knowledgeable comrade can help you cope with this task, school teacher or a tutor in mathematics and physics. Good luck with your physics studies!
Sergey Valerievich
At school, each of us came across a problem similar to the following. If a car moved part of the way at one speed, and the next part of the road at another, how to find the average speed?
What is this quantity and why is it needed? Let's try to figure this out.
Speed in physics is a quantity that describes the amount of distance traveled per unit of time. That is, when they say that a pedestrian’s speed is 5 km/h, this means that he covers a distance of 5 km in 1 hour.
The formula for finding speed looks like this:
V=S/t, where S is the distance traveled, t is time.
There is no single dimension in this formula, since it describes both extremely slow and very fast processes.
For example, an artificial Earth satellite travels about 8 km in 1 second, and the tectonic plates on which the continents are located, according to scientists’ measurements, diverge by only a few millimeters per year. Therefore, speed dimensions can be different - km/h, m/s, mm/s, etc.
The principle is that the distance is divided by the time required to cover the path. Do not forget about dimensionality if complex calculations are carried out.
In order not to get confused and not make a mistake in the answer, all quantities are given in the same units of measurement. If the length of the path is indicated in kilometers, and some part of it in centimeters, then until we get unity in dimension, we will not know the correct answer.
Constant speed
Description of the formula.
The simplest case in physics is uniform motion. The speed is constant and does not change throughout the entire journey. There are even speed constants tabulated—unchangeable values. For example, sound travels in air at a speed of 340.3 m/s.
And light is the absolute champion in this regard, it has the highest speed in our Universe - 300,000 km/s. These quantities do not change from the starting point of movement to the final point. They depend only on the medium in which they move (air, vacuum, water, etc.).
Uniform motion often occurs to us in everyday life. This is how a conveyor belt works in a plant or factory, a cable car on mountain roads, an elevator (except for very short periods of start and stop).
The graph of such a movement is very simple and represents a straight line. 1 second - 1 m, 2 seconds - 2 m, 100 seconds - 100 m. All points are on the same straight line.
Uneven speed
Unfortunately, it is extremely rare for things to be this ideal both in life and in physics. Many processes occur at an uneven speed, sometimes speeding up, sometimes slowing down.
Let's imagine the movement of a regular intercity bus. At the beginning of the journey, he accelerates, slows down at traffic lights, or even stops altogether. Then it goes faster outside the city, but slower on the ascents, and accelerates again on the descents.
If you depict this process in the form of a graph, you will get a very intricate line. You can determine the speed from the graph only for a specific point, but general principle No.
You will need a whole set of formulas, each of which is suitable only for its own section of the drawing. But there's nothing scary. To describe the movement of the bus, an average value is used.
You can find the average speed using the same formula. Indeed, we know the distance between bus stations and travel time has been measured. Divide one by the other and find the required value.
What is this for?
Such calculations are useful to everyone. We plan our day and movements all the time. Having a dacha outside the city, it makes sense to find out the average ground speed when traveling there.
This will make planning your weekend easier. Having learned to find this value, we can be more punctual and stop being late.
Let's return to the example proposed at the very beginning, when a car drove part of the way at one speed, and the other at a different speed. This type of problem is very often used in school curriculum. Therefore, when your child asks you to help him with a similar issue, it will be easy for you to do it.
By adding up the lengths of the path sections, you get the total distance. By dividing their values by the speeds indicated in the initial data, you can determine the time spent on each section. Adding them up, we get the time spent on the entire journey.
Mechanical movement of a body is the change in its position in space relative to other bodies over time. In this case, the bodies interact according to the laws of mechanics.
Section of mechanics describing geometric properties movement without taking into account the reasons causing it is called kinematics.
In a more general sense, motion is any spatial or temporal change in the state of a physical system. For example, we can talk about the movement of a wave in a medium.
Relativity of motion
Relativity is the dependence of the mechanical movement of a body on the reference system. Without specifying the reference system, it makes no sense to talk about motion.
Trajectory material point - a line in three-dimensional space, representing a set of points at which a material point was, is, or will be located when moving in space. It is important that the concept of a trajectory has a physical meaning even in the absence of any movement along it. In addition, even if there is an object moving along it, the trajectory itself cannot give anything regarding the causes of movement, that is, about the acting forces.
Path- the length of the section of the trajectory of a material point traversed by it in a certain time.
Speed(often denoted from the English velocity or French vitesse) is a vector physical quantity that characterizes the speed of movement and direction of movement of a material point in space relative to the selected reference system (for example, angular velocity). The same word can be called scalar quantity, more precisely, the modulus of the derivative of the radius vector.
Science also uses speed in in a broad sense, as the speed of change of some quantity (not necessarily the radius vector) depending on another (usually changes in time, but also in space or any other). For example, they talk about the rate of temperature change, the rate chemical reaction, group speed, connection speed, angular speed, etc. Mathematically characterized by the derivative of the function.
Speed units
Meter per second, (m/s), SI derived unit
Kilometer per hour, (km/h)
knot (nautical miles per hour)
The Mach number, Mach 1, is equal to the speed of sound in a given medium; Max n is n times faster.
How the unit depends on specific environmental conditions must be further defined.
The speed of light in a vacuum (denoted c)
In modern mechanics, the movement of a body is divided into types, and there is the following classification of types of body movement:
Translational motion in which any straight line associated with the body remains parallel to itself while moving
Rotational motion or rotation of a body around its axis, which is considered stationary.
Complex body movement consisting of translational and rotational movements.
Each of these types can be uneven and uniform (with non-constant and constant speed, respectively).
Average speed uneven movement
Average ground speed is the ratio of the length of the path traveled by the body to the time during which this path was covered:
Average ground speed, unlike instantaneous speed, is not a vector quantity.
The average speed is equal to the arithmetic mean of the speeds of the body during movement only in the case when the body moved at these speeds for the same periods of time.
At the same time, if, for example, the car moved half the way at a speed of 180 km/h, and the second half at a speed of 20 km/h, then the average speed will be 36 km/h. In examples like this, the average speed is equal to the harmonic mean of all speeds on individual, equal sections of the path.
Average moving speed
You can also enter the average speed for the movement, which will be a vector equal to the ratio of the movement to the time during which it was completed:
The average speed determined in this way can be equal to zero even if the point (body) actually moved (but at the end of the time interval returned to its original position).
If the movement occurred in a straight line (and in one direction), then the average ground speed is equal to the module of the average speed along the movement.
Rectilinear uniform motion- this is a movement in which a body (point) makes identical movements over any equal periods of time. The velocity vector of a point remains unchanged, and its displacement is the product of the velocity vector and time:
If you send coordinate axis along the straight line along which the point moves, then the dependence of the point’s coordinates on time is linear: , where is the initial coordinate of the point, is the projection of the velocity vector onto the x coordinate axis.
A point considered in an inertial reference system is in a state of uniform rectilinear movement, if the resultant of all forces applied to a point is equal to zero.
Rotational movement- type of mechanical movement. During rotational motion, absolutely solid its points describe circles located in parallel planes. The centers of all circles lie on the same straight line, perpendicular to the planes of the circles and called the axis of rotation. The axis of rotation can be located inside the body or outside it. The axis of rotation in a given reference system can be either movable or stationary. For example, in the reference frame associated with the Earth, the axis of rotation of the generator rotor at a power plant is stationary.
Characteristics of body rotation
With uniform rotation (N revolutions per second),
Rotational speed- number of body revolutions per unit time,
Rotation period- time of one full revolution. The rotation period T and its frequency v are related by the relation T = 1 / v.
Linear speed point located at a distance R from the axis of rotation
,
Angular velocity
body rotation.
Kinetic energy rotational movement
Where Iz- moment of inertia of the body relative to the axis of rotation. w - angular velocity.
Harmonic oscillator(in classical mechanics) is a system that, when displaced from an equilibrium position, experiences a restoring force proportional to the displacement.
If the restoring force is the only force acting on the system, then the system is called a simple or conservative harmonic oscillator. Free oscillations of such a system represent periodic movement around the equilibrium position (harmonic oscillations). The frequency and amplitude are constant, and the frequency does not depend on the amplitude.
If there is also a frictional force (damping) proportional to the speed of movement (viscous friction), then such a system is called a damped or dissipative oscillator. If the friction is not too great, then the system performs almost periodic motion - sinusoidal oscillations with a constant frequency and exponentially decreasing amplitude. The frequency of free oscillations of a damped oscillator turns out to be somewhat lower than that of a similar oscillator without friction.
If the oscillator is left to its own devices, it is said to oscillate freely. If present external force(time-dependent), then they say that the oscillator experiences forced oscillations.
Mechanical examples of a harmonic oscillator are a mathematical pendulum (with small angles of displacement), a mass on a spring, a torsion pendulum, and acoustic systems. Among other analogues of a harmonic oscillator, it is worth highlighting the electric harmonic oscillator (see LC circuit).
Sound, in a broad sense, are elastic waves that propagate longitudinally in a medium and create mechanical vibrations in it; in a narrow sense, the subjective perception of these vibrations by the special sense organs of animals or humans.
Like any wave, sound is characterized by amplitude and frequency spectrum. Typically, a person hears sounds transmitted through the air in the frequency range from 16 Hz to 20 kHz. Sound below the range of human audibility is called infrasound; higher: up to 1 GHz - ultrasound, more than 1 GHz - hypersound. Among the sounds heard, one should also highlight phonetic, speech sounds and phonemes (which make up oral speech) and musical sounds(of which music consists).
Physical parameters of sound
Oscillatory speed- a value equal to the product of the oscillation amplitude A particles of the medium through which a periodic sound wave passes, at the angular frequency w:
where B is the adiabatic compressibility of the medium; p - density.
Like light waves, sound waves can also be reflected, refracted, etc.
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Remember that speed is given by both a numerical value and a direction. Velocity describes how quickly a body's position changes, as well as the direction in which that body is moving. For example, 100 m/s (south).
Find the total displacement, that is, the distance and direction between the starting and ending points of the path. As an example, consider a body moving with constant speed in one direction.
- For example, a rocket was launched in a northerly direction and moved for 5 minutes at a constant speed of 120 meters per minute. To calculate the total displacement, use the formula s = vt: (5 minutes) (120 m/min) = 600 m (north).
- If the problem is given a constant acceleration, use the formula s = vt + ½at 2 (the next section describes a simplified way to work with constant acceleration).
Find the total travel time. In our example, the rocket travels for 5 minutes. Average speed can be expressed in any unit of measurement, but in the International System of Units, speed is measured in meters per second (m/s). Convert minutes to seconds: (5 minutes) x (60 seconds/minute) = 300 seconds.
- Even if in a scientific problem time is given in hours or other units of measurement, it is better to first calculate the speed and then convert it to m/s.
Calculate the average speed. If you know the value of the displacement and the total travel time, you can calculate the average speed using the formula v av = Δs/Δt. In our example, the average speed of the rocket is 600 m (north) / (300 seconds) = 2 m/s (north).
- Be sure to indicate the direction of travel (for example, “forward” or “north”).
- In the formula v av = Δs/Δt the symbol "delta" (Δ) means "change in magnitude", that is, Δs/Δt means "change in position to change in time".
- The average speed can be written as v av or as v with a horizontal bar on top.
Solution more complex tasks, for example, if the body rotates or the acceleration is not constant. In these cases, the average speed is still calculated as the ratio of total displacement to total time. It doesn't matter what happens to the body between the starting and ending points of the path. Here are some examples of problems with the same total displacement and total time (and therefore the same average speed).
- Anna walks west at 1 m/s for 2 seconds, then instantly accelerates to 3 m/s and continues to walk west for 2 seconds. Its total displacement is (1 m/s)(2 s) + (3 m/s)(2 s) = 8 m (to the west). Total time on the way: 2 s + 2 s = 4 s. Her average speed: 8 m / 4 s = 2 m/s (west).
- Boris walks west at 5 m/s for 3 seconds, then turns around and walks east at 7 m/s for 1 second. We can consider the movement to the east as a "negative movement" to the west, so the total movement is (5 m/s)(3 s) + (-7 m/s)(1 s) = 8 meters. The total time is 4 s. Average speed is 8 m (west) / 4 s = 2 m/s (west).
- Julia walks 1 meter north, then walks 8 meters west, and then walks 1 meter south. The total travel time is 4 seconds. Draw a diagram of this movement on paper and you will see that it ends 8 meters west of the starting point, so the total movement is 8 m. The total travel time was 4 seconds. Average speed is 8 m (west) / 4 s = 2 m/s (west).
To calculate your average speed, use a simple formula: Speed = Distance traveled Time (\displaystyle (\text(Speed))=(\frac (\text(Distance traveled))(\text(Time)))). But in some problems two speed values are given - on different sections of the path traveled or at different time intervals. In these cases, you need to use other formulas to calculate the average speed. Skills in solving such problems can be useful in real life, and the problems themselves may appear in exams, so remember the formulas and understand the principles of solving problems.
Steps
One path value and one time value
- the length of the path traveled by the body;
- the time it took the body to travel this path.
- For example: a car traveled 150 km in 3 hours. Find the average speed of the car.
-
Formula: , where v (\displaystyle v)- average speed, s (\displaystyle s)- the distance traveled, t (\displaystyle t)- the time it took to travel the path.
Substitute the distance traveled into the formula. Substitute the path value instead s (\displaystyle s).
- In our example, the car traveled 150 km. The formula will be written like this: v = 150 t (\displaystyle v=(\frac (150)(t))).
-
Substitute time into the formula. Substitute the time value instead t (\displaystyle t).
- In our example, the car drove for 3 hours. The formula will be written like this: .
-
Divide the journey by time. You will find the average speed (usually measured in kilometers per hour).
- In our example:
v = 150 3 (\displaystyle v=(\frac (150)(3)))
Thus, if a car traveled 150 km in 3 hours, then it moved at an average speed of 50 km/h.
- In our example:
-
Calculate the total distance traveled. To do this, add up the values of the traveled sections of the path. Substitute the total distance traveled into the formula (instead of s (\displaystyle s)).
- In our example, the car drove 150 km, 120 km and 70 km. Total distance traveled: .
-
T (\displaystyle t)).
- . Thus, the formula will be written like this: .
-
- In our example:
v = 340 6 (\displaystyle v=(\frac (340)(6)))
Thus, if a car traveled 150 km in 3 hours, 120 km in 2 hours, 70 km in 1 hour, then it moved at an average speed of 57 km/h (rounded).
- In our example:
For several speed values and several time values
-
Look at these values. Use this method if the following quantities are given:
Write down the formula to calculate the average speed. Formula: v = s t (\displaystyle v=(\frac (s)(t))), Where v (\displaystyle v)- average speed, s (\displaystyle s)- total distance traveled, t (\displaystyle t)- the total time during which the path was covered.
-
Calculate common path. To do this, multiply each speed by the corresponding time. This way you will find the length of each section of the path. To calculate the total path, add up the values of the traveled sections of the path. Substitute the total distance traveled into the formula (instead of s (\displaystyle s)).
- For example:
50 km/h for 3 hours = 50 × 3 = 150 (\displaystyle 50\times 3=150) km
60 km/h for 2 hours = 60 × 2 = 120 (\displaystyle 60\times 2=120) km
70 km/h for 1 hour = 70 × 1 = 70 (\displaystyle 70\times 1=70) km
Total distance traveled: 150 + 120 + 70 = 340 (\displaystyle 150+120+70=340) km. Thus, the formula will be written like this: v = 340 t (\displaystyle v=(\frac (340)(t))).
- For example:
-
Calculate the total travel time. To do this, add up the times it took to cover each section of the path. Substitute the total time into the formula (instead of t (\displaystyle t)).
- In our example, the car drove for 3 hours, 2 hours and 1 hour. Total travel time: 3 + 2 + 1 = 6 (\displaystyle 3+2+1=6). Thus, the formula will be written like this: v = 340 6 (\displaystyle v=(\frac (340)(6))).
-
Divide the total distance by the total time. You will find the average speed.
- In our example:
v = 340 6 (\displaystyle v=(\frac (340)(6)))
v = 56, 67 (\displaystyle v=56,67)
Thus, if a car was moving at a speed of 50 km/h for 3 hours, at a speed of 60 km/h for 2 hours, at a speed of 70 km/h for 1 hour, then it was moving at an average speed of 57 km/h ( rounded).
- In our example:
For two speed values and two identical time values
-
Look at these values. Use this method if the following quantities and conditions are given:
- two or more values of the speeds at which the body moved;
- the body moved at certain speeds for equal periods of time.
- For example: a car moved at a speed of 40 km/h for 2 hours and at a speed of 60 km/h for another 2 hours. Find the average speed of the car along the entire journey.
-
Write down a formula to calculate the average speed if given two speeds at which a body moves for equal periods of time. Formula: v = a + b 2 (\displaystyle v=(\frac (a+b)(2))), Where v (\displaystyle v)- average speed, a (\displaystyle a)- the speed of the body during the first period of time, b (\displaystyle b)- the speed of the body during the second (same as the first) period of time.
- In such problems, the values of time intervals are not important - the main thing is that they are equal.
- If several speed values and equal time intervals are given, rewrite the formula as follows: v = a + b + c 3 (\displaystyle v=(\frac (a+b+c)(3))) or v = a + b + c + d 4 (\displaystyle v=(\frac (a+b+c+d)(4))). If the time intervals are equal, add up all the speed values and divide them by the number of such values.
-
Substitute the speed values into the formula. It doesn't matter what value to substitute a (\displaystyle a), and which one - instead b (\displaystyle b).
- For example, if the first speed is 40 km/h and the second speed is 60 km/h, the formula will be written as follows: .
-
Add the two speeds together. Then divide the amount by two. You will find the average speed along the entire path.
- For example:
v = 40 + 60 2 (\displaystyle v=(\frac (40+60)(2)))
v = 100 2 (\displaystyle v=(\frac (100)(2)))
v = 50 (\displaystyle v=50)
Thus, if a car moved at a speed of 40 km/h for 2 hours and at a speed of 60 km/h for another 2 hours, the average speed of the car along the entire journey was 50 km/h.
- For example: