Average travel speed formula. What is the formula for calculating average speed?

Medium speed tasks (hereinafter referred to as SV). We have already looked at tasks involving linear motion. I recommend looking at the articles "" and "". Typical tasks for average speed are a group of movement problems, they are included in the Unified State Examination in mathematics, and such a task may very likely appear in front of you at the time of the exam itself. The problems are simple and can be solved quickly.

The idea is this: imagine an object of movement, such as a car. He travels certain sections of the path at different speeds. The entire journey takes a certain amount of time. So here it is: average speed this is the constant speed at which the car would cover a given distance in the same time. That is, the formula for average speed is:

If there were two sections of the path, then

If three, then accordingly:

*In the denominator we sum up the time, and in the numerator the distances traveled during 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 third at a speed of 45 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

As already said, it is necessary to divide the entire path into the entire time of movement. The condition says about three sections of the path. Formula:

Let us denote the whole by S. Then the car drove the first third of the way:

The car drove the second third of the way:

The car drove the last third of the way:

Thus

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 third at a speed of 110 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

The car drove for the first hour at a speed of 100 km/h, for the next two hours at a speed of 90 km/h, and then for two hours at a speed of 80 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

The condition says about three sections of the path. We will search for the 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 car drove at a speed of 50 km/h for the first two hours, at a speed of 100 km/h for the next hour, and at a speed of 75 km/h for two hours. Find the vehicle's IC along the entire route. Give your answer in km/h.

The car drove for the first 120 km at a speed of 60 km/h, for the next 120 km at a speed of 80 km/h, and then for 150 km at a speed of 100 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

It is said about three sections of the path. Formula:

The length of the sections is given. Let us 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 car drove for the first 190 km 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 IC along the entire route. Give your answer in km/h.

Half the time spent on the road, the car was traveling 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 IC along the entire route. 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 don't see it, then register on the site! The registration (login) button is located in the MAIN MENU of the site. After registration, log in to the site and refresh this page.

The traveler crossed the sea on a yacht with average speed 17 km/h. He flew back on a sports plane at a speed of 323 km/h. Find the average speed of the traveler along the entire journey. Give your answer in km/h.

Best regards, Alexander.

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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.

1. 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))).

2. 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: .

3. 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.

4. 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: .

5. T (\displaystyle t)).

   • . Thus, the formula will be written like this: .
6. • 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).

For several speed values ​​and several time values

1. 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 traveled.

2. 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))).

3. 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 time on the way: 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))).

4. Divide the total 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 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).

For two speed values ​​and two identical time values

1. 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.

2. 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.

3. 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 like this: .

4. 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.

Instructions

Consider the function f(x) = |x|. To begin with, this is an unsigned modulus, that is, the graph of the function g(x) = x. This graph is a straight line passing through the origin and the angle between this straight line and the positive direction of the x-axis is 45 degrees.

Since the modulus is a non-negative quantity, the part that is below the abscissa axis must be mirrored relative to it. For the function g(x) = x, we find that the graph after such a mapping will look like V. This new graph will be a graphical interpretation of the function f(x) = |x|.

Video on the topic

Please note

The modulus graph of a function will never be in the 3rd and 4th quarters, since the modulus cannot accept negative values.

Useful advice

If a function contains several modules, then they need to be expanded sequentially and then stacked on top of each other. The result will be the desired graph.

Sources:

• how to graph a function with modules

Kinematics problems in which you need to calculate speed, time or the path of uniformly and rectilinearly moving bodies that meet in school course algebra and physics. To solve them, find in the condition quantities that can be equalized. If the condition requires defining time at a known speed, use the following instructions.

You will need

• - pen;
• - paper for notes.

Instructions

The simplest case is the movement of one body with a given uniform speed Yu. The distance that the body has traveled is known. Find on the way: t = S/v, hour, where S is the distance, v is the average speed bodies.

The second is for oncoming movement of bodies. A car moves from point A to point B with speed 50 km/h. A moped with a speed 30 km/h. The distance between points A and B is 100 km. Need to find time through which they will meet.

Label the meeting point K. Let the distance AK of the car be x km. Then the motorcyclist’s path will be 100 km. From the problem conditions it follows that time On the road, a car and a moped have the same experience. Make up the equation: x/v = (S-x)/v’, where v, v’ – and the moped. Substituting the data, solve the equation: x = 62.5 km. Now time: t = 62.5/50 = 1.25 hours or 1 hour 15 minutes.

Third example - the same conditions are given, but the car left 20 minutes later than the moped. Determine how long the car will travel before meeting the moped.

Create an equation similar to the previous one. But in this case time a moped's travel time will be 20 minutes faster than that of a car. To equalize the parts, subtract one third of an hour from the right side of the expression: x/v = (S-x)/v’-1/3. Find x – 56.25. Calculate time: t = 56.25/50 = 1.125 hours or 1 hour 7 minutes 30 seconds.

The fourth example is a problem involving the movement of bodies in one direction. A car and a moped are moving from point A at the same speeds. It is known that the car left half an hour later. After what time will he catch up with the moped?

In this case, the distance traveled will be the same vehicles. Let time the car will travel x hours, then time the moped's journey will be x+0.5 hours. You have the equation: vx = v’(x+0.5). Solve the equation by substituting , and find x – 0.75 hours or 45 minutes.

Fifth example – a car and a moped are moving at the same speeds in the same direction, but the moped left point B, located 10 km from point A, half an hour earlier. Calculate after what time After the start, the car will catch up with the moped.

The distance traveled by the car is 10 km more. Add this difference to the motorcyclist’s path and equalize the parts of the expression: vx = v’(x+0.5)-10. Substituting the speed values ​​and solving it, you get: t = 1.25 hours or 1 hour 15 minutes.

Sources:

• what is the speed of the time machine

Instructions

Calculate the average of a body moving uniformly along a section of path. Such speed is the easiest to calculate, since it does not change over the entire segment movement and equals the average. This can be expressed in the form: Vрд = Vср, where Vрд – speed uniform movement, and Vav – average speed.

Calculate the average speed uniformly slow (uniformly accelerated) movement in this area, for which it is necessary to add the initial and final speed. Divide the result by two, which

All tasks in which there is movement of objects, their movement or rotation, are somehow related to speed.

This term characterizes the movement of an object in space over a certain period of time - the number of units of distance per unit of time. He is a frequent “guest” of both sections of mathematics and physics. The original body can change its location both uniformly and with acceleration. In the first case, the speed value is static and does not change during movement, in the second, on the contrary, it increases or decreases.

How to find speed - uniform motion

If the speed of movement of the body remained unchanged from the beginning of the movement until the end of the path, then we are talking about movement with constant acceleration - uniform movement. It can be straight or curved. In the first case, the trajectory of the body is a straight line.

Then V=S/t, where:

• V – desired speed,
• S – distance traveled (total path),
• t – total movement time.

How to find speed - acceleration is constant

If an object was moving with acceleration, then its speed changed as it moved. In this case, the following expression will help you find the desired value:

V=V (start) + at, where:

• V (start) – the initial speed of the object,
• a – acceleration of the body,
• t – total travel time.

How to find speed - uneven motion

IN in this case There is a situation when the body passes through different sections of the path in different times.
S(1) – for t(1),
S(2) – for t(2), etc.

In the first section, the movement occurred at the “tempo” V(1), in the second – V(2), etc.

To find out the speed of movement of an object along the entire path (its average value), use the expression:

How to find speed - rotation of an object

In the case of rotation, we are talking about angular velocity, which determines the angle through which the element rotates per unit time. The desired value is indicated by the symbol ω (rad/s).

• ω = Δφ/Δt, where:

Δφ – angle passed (angle increment),
Δt – elapsed time (movement time – time increment).

• If the rotation is uniform, the desired value (ω) is associated with such a concept as the period of rotation - how long it will take for our object to make 1 full revolution. In this case:

ω = 2π/T, where:
π – constant ≈3.14,
T – period.

Or ω = 2πn, where:
π – constant ≈3.14,
n – circulation frequency.

• With a known linear speed object for each point on the path of movement and the radius of the circle along which it moves, to find the speed ω you will need the following expression:

ω = V/R, where:
V – numerical value vector quantity(linear speed),
R is the radius of the body’s trajectory.

How to find speed - moving points closer and further away

In problems of this kind, it would be appropriate to use the terms speed of approach and speed of distance.

If objects are directed towards each other, then the speed of approaching (removing) will be as follows:
V (closer) = V(1) + V(2), where V(1) and V(2) are the velocities of the corresponding objects.

If one of the bodies catches up with the other, then V (closer) = V(1) – V(2), V(1) is greater than V(2).

How to find speed - movement on a body of water

If events unfold on water, then the speed of the current (i.e., the movement of water relative to a stationary shore) is added to the object’s own speed (the movement of the body relative to the water). How are these concepts interrelated?

In the case of moving with the current, V=V(own) + V(flow).
If against the current – ​​V=V(own) – V(current).

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).