To calculate your average speed, use a simple formula: Speed = Distance traveled Time (\ displaystyle (\ text (Speed)) = (\ frac (\ text (Distance)) (\ text (Time))))... But in some problems, two values of speed are given - at different sections of the distance traveled or at different periods of time. In these cases, you need to use other formulas to calculate the average speed. The skills of solving such problems can be useful in real life, and the problems themselves can be found 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 traversed by the body;
- the time during which the body has traveled this path.
- For example: the car traveled 150 km in 3 hours. Find the average speed of the car.
-
Formula:, where v (\ displaystyle v)- average speed, s (\ displaystyle s)- distance traveled, t (\ displaystyle t)- the time during which the path was covered.
Substitute the path traveled into the formula. Substitute the path value for s (\ displaystyle s).
- In our example, the car has traveled 150 km. The formula will be written like this: v = 150 t (\ displaystyle v = (\ frac (150) (t))).
-
Plug the time into the formula. Substitute the time value for t (\ displaystyle t).
- In our example, the car has been driving for 3 hours. The formula will be written as follows:.
-
Divide the path for 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 was moving 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 traveled 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 the car traveled 150 km in 3 hours, 120 km in 2 hours, 70 km in 1 hour, then it was moving at an average speed of 57 km / h (rounded off).
- In our example:
For several values of speeds and several values of time
-
Look at the given values. Use this method if the following values are given:
Write down the formula for calculating 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 for which the path was covered.
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Calculate the common path. To do this, multiply each speed by the corresponding time. This will give you the length of each section of the path. Add up the distance traveled to calculate the total path. Substitute the total distance traveled into the formula (instead of s (\ displaystyle s)).
- For instance:
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 instance:
-
Calculate the total travel time. To do this, add up the times for which each section of the path was covered. Substitute the total time in 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 shared path 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 the 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 moved at an average speed of 57 km / h ( rounded).
- In our example:
For two values of speeds and two identical values of time
-
Look at the given values. Use this method if the following values and conditions are given:
- two or more values of the velocities with which the body was moving;
- the body moved at certain speeds for regular intervals of time.
- For example: the car was moving at 40 km / h for 2 hours and at 60 km / h for another 2 hours. Find the average speed of the car along the way.
-
Write down the formula for calculating the average speed if you are given two speeds with which the body moves during equal periods of time. Formula: v = a + b 2 (\ displaystyle v = (\ frac (a + b) (2))), where v (\ displaystyle v)- average speed, a (\ displaystyle a)- body speed during the first period of time, b (\ displaystyle b)- the speed of the body during the second (the same as the first) period of time.
- In such tasks, the values of the time intervals are not important - the main thing is that they are equal.
- If you are given several speeds and equal intervals of time, 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 velocities and divide them by the number of such values.
-
Plug in the speed values into the formula. It doesn't matter what value you substitute for a (\ displaystyle a), and which - instead of 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 like this:.
-
Add the two speeds together. Then divide the sum by two. You will find the average speed all along the way.
- For instance:
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 the car was moving at 40 km / h for 2 hours and at 60 km / h for another 2 hours, the average speed of the car along the way was 50 km / h.
- For instance:
Mechanical movement body is called 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.
The section of mechanics describing the geometric properties of motion without taking into account the reasons that cause it is called kinematics.
In a more general sense, motion refers to 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.
Motion relativity
Relativity - the dependence of the mechanical motion of a body on the frame of reference Without specifying the frame of reference, it makes no sense to talk about motion.
Material point trajectory- a line in three-dimensional space, which is a set of points at which a material point was, is or will be when it moves in space. It is essential that the concept of a trajectory has a physical meaning even in the absence of any movement along it. In addition, even in the presence of an object moving along it, the trajectory itself cannot give anything about the reasons for the movement, that is, about the acting forces.
Path- the length of the section of the trajectory of a material point, traversed by it for a certain time.
Speed(often denoted from English velocity or French vitesse) is a vector physical quantity that characterizes the speed and direction of movement of a material point in space relative to the selected frame of reference (for example, angular velocity). The same word can be called a scalar quantity, more precisely, the modulus of the derivative of the radius vector.
In science, speed is also used in a broad sense, as the rate of change of any quantity (not necessarily the radius vector) depending on another (more often changes in time, but also in space or any other). So, for example, they talk about the rate of temperature change, the rate of a chemical reaction, the group rate, the rate of connection, angular velocity, etc. The derivative of the function is mathematically characterized.
Speed units
Meter per second, (m / s), SI derived unit
Kilometer per hour, (km / h)
knot (nautical mile per hour)
Mach number, Mach 1 is equal to the speed of sound in a given environment; Max n is n times faster.
As a unit, depending on the specific conditions of the environment, it should be additionally 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 body movement types:
Translational movement, in which any straight line associated with the body remains parallel to itself when moving
Rotational movement or rotation of a body around its axis, which is considered motionless.
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 of uneven movement
Average ground speed is the ratio of the length of the path traversed by the body to the time during which this path was traversed:
The average ground speed, in contrast to the instantaneous speed, is not a vector quantity.
The average speed is equal to the arithmetic mean of the speed of the body during movement only if the body was moving at these speeds for the same time intervals.
At the same time, if, for example, half of the way the car was moving 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 average of all speeds on separate, equal sections of the path.
Average travel speed
You can also enter the average speed along the movement, which will be a vector equal to the ratio of the movement to the time it took:
The average speed determined in this way can be zero even if the point (body) actually moved (but at the end of the time interval returned to its original position).
If the movement took place in a straight line (and in one direction), then the average ground speed is equal to the modulus of the average speed along the movement.
Rectilinear uniform movement- this is a movement in which the body (point) for any equal intervals of time makes the same movements. The velocity vector of a point remains unchanged, and its movement is the product of the velocity vector by time:
If we direct the coordinate axis along the straight line along which the point moves, then the dependence of the coordinate of the point on time is linear:, where is the initial coordinate of the point, is the projection of the velocity vector onto the coordinate axis x.
A point considered in an inertial reference system is in a state of uniform rectilinear motion if the resultant of all forces applied to the point is zero.
Rotational motion- type of mechanical movement. During the rotational motion of an absolutely rigid body, its points describe circles located in parallel planes. The centers of all circles lie on one 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 and outside it. The axis of rotation in a given frame of reference can be both movable and fixed. For example, in a frame of reference related to the Earth, the axis of rotation of the rotor of a generator at a power plant is stationary.
Body rotation characteristics
With uniform rotation (N revolutions per second),
Rotation frequency- the number of body revolutions per unit of time,
Rotation period- the time of one complete revolution. The period of rotation T and its frequency v are related by the ratio T = 1 / v.
Linear Velocity point located at a distance R from the axis of rotation
,
Angular velocity body rotation.
Kinetic energy rotary motion
Where I z- moment of inertia of the body about the axis of rotation. w - angular velocity.
Harmonic Oscillator(in classical mechanics) is a system that, when displaced from an equilibrium position, experiences the action of 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 vibrations of such a system are periodic motion around the equilibrium position (harmonic vibrations). In this case, 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 motion (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 itself, then they say that it makes free oscillations. If there is an external force (depending on time), then the oscillator is said to experience forced oscillations.
Mechanical examples of a harmonic oscillator are a mathematical pendulum (with small displacement angles), a spring weight, a torsion pendulum, and acoustic systems. Among other analogs of the harmonic oscillator, it is worth highlighting the electrical harmonic oscillator (see LC circuit).
Sound, in a broad sense - elastic waves propagating longitudinally in a medium and creating mechanical vibrations in it; in a narrow sense - the subjective perception of these vibrations by the special senses of animals or humans.
Like any wave, sound is characterized by amplitude and frequency spectrum. Usually, a person hears sounds transmitted through the air in the frequency range from 16 Hz to 20 kHz. Sound below the human hearing range is called infrasound; higher: up to 1 GHz - by ultrasound, more than 1 GHz - by hypersound. Among the audible sounds, phonetic, speech sounds and phonemes (of which oral speech consists) and musical sounds (of which music consists) should also be highlighted.
Physical parameters of sound
Oscillatory speed- a value equal to the product of the vibration amplitude A particles of the medium through which a periodic sound wave passes, to the angular frequency w:
where B is the adiabatic compressibility of the medium; p is the density.
Like light waves, sound waves can also be reflected, refracted, etc.
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Medium speed tasks (hereinafter referred to as SK). We have already considered the tasks for rectilinear movement. I recommend looking at the articles "" and "". Typical tasks for an average speed are a group of tasks for movement, they are included in the exam in mathematics, and such a task may very likely be in front of you at the time of the exam itself. The tasks are simple, they are solved quickly.
The point is this: imagine an object of movement, for example a car. He goes through certain sections of the path at different speeds. The whole journey takes a certain amount of time. So: the average speed is such a constant speed with which the car would cover the given path during the same time.That is, the formula for the average speed is as follows:
If there were two sections of the path, then
If three, then, respectively:
* In the denominator, we summarize the time, and in the numerator, the distances traveled in the corresponding time intervals.
The car drove the first third of the route at a speed of 90 km / h, the second third - at a speed of 60 km / h, and the last - at a speed of 45 km / h. Find the vehicle's SK along the way. Give your answer in km / h.
As already mentioned, it is necessary to divide the entire path by the entire movement time. The condition says about three sections of the path. Formula:
Let us denote the whole let S. Then the car drove the first third of the way:
The car was driving the second third of the way:
The car was driving the last third of the way:
In this way
Decide for yourself:
The car drove the first third of the route at a speed of 60 km / h, the second third - at a speed of 120 km / h, and the last - at a speed of 110 km / h. Find the vehicle's SK along the way. Give your answer in km / h.
The first hour the car drove at a speed of 100 km / h, the next two hours - at a speed of 90 km / h, and then two hours - at a speed of 80 km / h. Find the vehicle's SK along the way. Give your answer in km / h.
The condition says about three sections of the path. We will search for SC using the formula:
The sections of the path are not given to us, but we can easily calculate them:
The first section of the route was 1 ∙ 100 = 100 kilometers.
The second section of the route was 2 ∙ 90 = 180 kilometers.
The third section of the route was 2 ∙ 80 = 160 kilometers.
We calculate the speed:
Decide for yourself:
The first two hours the car drove at a speed of 50 km / h, the next hour - at a speed of 100 km / h, and then two hours - at a speed of 75 km / h. Find the vehicle's SK along the way. Give your answer in km / h.
The first 120 km the car drove at a speed of 60 km / h, the next 120 km - at a speed of 80 km / h, and then 150 km - at a speed of 100 km / h. Find the vehicle's SK along the way. Give your answer in km / h.
It is said about three sections of the path. Formula:
The length of the sections is given. Let's determine the time that the car spent on each section: 120/60 hours were spent on the first section, 120/80 hours on the second section, 150/100 hours on the third. We calculate the speed:
Decide for yourself:
The first 190 km the car drove at a speed of 50 km / h, the next 180 km - at a speed of 90 km / h, and then 170 km - at a speed of 100 km / h. Find the vehicle's SK along the way. Give your answer in km / h.
Half of the time spent on the road, the car drove at a speed of 74 km / h, and the second half of the time - at a speed of 66 km / h. Find the vehicle's SK along the way. Give your answer in km / h.
* There is a problem about a traveler who crossed the sea. The guys have problems with the solution. If you do not see it, then register on the site! The registration (login) button is located in the MAIN MENU of the site. After registration, enter the site and refresh this page.
The traveler swam across the sea on a yacht with average speed 17 km / h He flew back in a sports plane at a speed of 323 km / h. Find the average speed of the traveler along the way. Give your answer in km / h.
Sincerely, Alexander.
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Remember that speed is given by both a numerical value and a direction. Velocity describes the rate at which a body’s position changes, as well as the direction in which the body is moving. For example, 100 m / s (south).
Find the total displacement, that is, the distance and direction between the start and end points of the path. As an example, consider a body moving at a constant speed in one direction.
- For example, the 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 constant acceleration, use the formula s = vt + ½at 2 (the next section describes a simplified way of working 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 the 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 displacement and the total travel time, you can calculate the average speed using the formula v cf = Δs / Δt. In our example, the average rocket speed is 600 m (north) / (300 seconds) = 2 m / s (north).
- Do not forget to indicate the direction of travel (for example, "forward" or "north").
- In the formula v cf = Δs / Δt the symbol "delta" (Δ) means "change in value", that is, Δs / Δt means "change in position to change in time".
- Average speed can be written as v avg or v with a horizontal bar at the top.
Solving more complex problems, for example, if the body is rotating or the acceleration is not constant. In these cases, the average speed is still calculated as the ratio of the total travel to the total time. It doesn't matter what happens to the body between the start and end points of the path. Here are some examples of tasks with the same total travel 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 west for 2 seconds. Its total movement is (1 m / s) (2 s) + (3 m / s) (2 s) = 8 m (to the west). Total travel time: 2 s + 2 s = 4 s. Its 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 eastward movement as "negative movement" westward, 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 to the north, then walks 8 meters to the west, and then goes 1 meter to the 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, that is, the total movement is 8 meters. The total travel time was 4 seconds. Average speed is 8 m (west) / 4 s = 2 m / s (west).
The concept of speed is one of the main concepts in kinematics.
Many people probably know that speed is a physical quantity that shows how fast (or how slowly) a moving body moves in space. Of course, we are talking about movement in the selected frame of reference. Did you know, however, that not one, but three concepts of speed are used? There is a speed at a given moment in time, called instantaneous speed, and there are two concepts of the average speed over a given period of time - the average ground speed (in English speed) and the average speed of movement (in English velocity).
We will consider a material point in the coordinate system x, y, z(fig. a).
Position A points at time t characterized by coordinates x (t), y (t), z (t) representing the three components of the radius vector ( t). The point moves, its position in the selected coordinate system changes over time - the end of the radius vector ( t) describes a curve called the trajectory of the moving point.
The trajectory described for the time interval from t before t + Δt, is shown in Figure b. 
Across B the position of the point at the moment t + Δt(it is fixed by the radius vector ( t + Δt)). Let Δs- the length of the considered curvilinear trajectory, i.e. the path traversed by the point during the time from t before t + Δt.
The average ground speed of a point for a given period of time is determined by the ratio 
It's obvious that v p- scalar value; it is only characterized by a numerical value.
The vector shown in figure b
is called the displacement of a material point for a time from t before t + Δt.
The average speed of movement for a given period of time is determined by the ratio 
It's obvious that v cf Is a vector quantity. Vector direction v cf coincides with the direction of travel Δr.
Note that in the case of rectilinear movement, the average ground speed of the moving point coincides with the module of the average speed of movement.
The motion of a point along a rectilinear or curvilinear trajectory is called uniform if in relation (1) the value of vp does not depend on Δt... If, for example, we reduce Δt 2 times, then the length of the path traveled by the point Δs will decrease by 2 times. With uniform motion, the point travels in equal intervals of path of equal length.
Question:
Is it possible to assume that with a uniform motion of a point from Δt also does not depend on the vector cp of the average speed along the displacement?
Answer:
This can be considered only in the case of rectilinear motion (in this case, recall, the modulus of the average speed of movement is equal to the average ground speed). If the uniform motion is performed along a curvilinear trajectory, then with a change in the averaging interval Δt both the modulus and the direction of the average velocity vector along the displacement will change. With uniform curvilinear movement at equal intervals of time Δt different displacement vectors will correspond Δr(and hence, different vectors v cf).
True, in the case of uniform movement along a circle, equal values of the displacement modulus will correspond to equal intervals of time | r |(and hence equal | v cf |). But the directions of displacements (and hence the vectors v cf) and in this case will be different for the same Δt... This can be seen in the figure, 
Where a point uniformly moving along a circle describes equal arcs in equal intervals of time AB, BC, CD... Although the displacement vectors 1
, 2
, 3
have the same modules, but their directions are different, so there is no need to talk about the equality of these vectors.
Note
Of the two average speeds in problems, the average ground speed is usually considered, and the average travel speed is used quite rarely. However, it deserves attention, since it allows us to introduce the concept of instantaneous velocity.