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Since ancient times, people have been bothered by the thought of achieving super speeds, just as they are haunted by thoughts about heights and flying machines. In fact, these are two very closely related concepts. How quickly you can get from one point to another on an aircraft in our time depends entirely on speed. Let us consider the methods and formulas for calculating this indicator, as well as time and distance.
- How to calculate speed?
- through the formula for finding power;
- through differential calculus;
by angular parameters and so on. This article discusses the simplest method with the simplest formula - finding the value of this parameter through distance and time. By the way, these indicators are also present in the differential calculation formulas.
- The formula looks like this:
- v - object speed,
- S is the distance that the object has traveled or must travel,
t is the time during which the distance has been or should be covered. As you can see, there is nothing complicated in the formula for the first grade of high school. By substituting the appropriate values instead of the letter designations, you can calculate the speed of movement of the object. For example, let’s find the speed of a car if it travels 100 km in 1 hour 30 minutes. First you need to convert 1 hour 30 minutes to hours
, since in most cases the unit of measurement of the parameter under consideration is considered to be kilometers per hour (km/h). So, 1 hour 30 minutes is equal to 1.5 hours, because 30 minutes is half or 1/2 or 0.5 hours. Adding together 1 hour and 0.5 hours we get 1.5 hours.
Now you need to substitute the existing values instead of alphabetic characters:
Here v=66.66 km/h, and this value is very approximate (for those who don’t know, it’s better to read about this in specialized literature), S=100 km, t=1.5 hours.
In this simple way you can find speed through time and distance.
So what to do, if you need to find the average value? In principle, the calculations shown above ultimately give the result of the average value of the parameter we are looking for. However, a more accurate value can be derived if it is known that in some areas the speed of the object was not constant compared to others. Then use this type of formula:
vav=(v1+v2+v3+...+vn)/n, where v1, v2, v3, vn are the values of the object’s speeds on individual sections of the path S, n is the number of these sections, vav is the average speed of the object throughout ways.
The same formula can be written differently, using the path and the time during which the object traveled this path:
- vav=(S1+S2+...+Sn)/t, where vav is the average speed of the object along the entire path,
- S1, S2, Sn - individual uneven sections of the entire path,
- t is the total time during which the object passed all sections.
You can also write to use this type of calculation:
- vср=S/(t1+t2+...+tn), where S is the total distance traveled,
- t1, t2, tn - time of passage of individual sections of distance S.
But you can write the same formula in a more precise version:
vср=S1/t1+S2/t2+...+Sn/tn, where S1/t1, S2/t2, Sn/tn are formulas for calculating the speed on each individual section of the entire path S.
Thus, it is very easy to find the desired parameter using the above formulas. They are very simple, and as already indicated, they are used in primary grades. More complex formulas are based on the same formulas and on the same principles of construction and calculation, but have a different, more complex form, more variables and different coefficients. This is necessary to obtain the most accurate indicator values.
Other calculation methods
There are other methods and methods that help calculate the values of the parameter in question. An example is the formula for calculating power:
N=F*v*cos α, where N is mechanical power,
v - speed,
cos α - cosine of the angle between the force and velocity vectors.
Methods for calculating distance and time
Conversely, knowing the speed, you can find the value of distance or time. For example:
S=v*t, where v is clear what it is,
S is the distance to be found,
t is the time it took the object to travel this distance.
This way the distance value is calculated.
Or calculate the time value, for which the distance has been traveled:
t=S/v, where v is the same speed,
S - distance, distance traveled,
t is the time whose value in this case needs to be found.
To find the average values of these parameters, there are quite a few representations of both this formula and all others. The main thing is to know the basic rules of permutations and calculations. And it’s even more important to know the formulas themselves, and better by heart. If you can’t remember, then it’s better to write it down. This will help, no doubt about it.
Using such permutations, you can easily find time, distance and other parameters using the necessary, correct methods for calculating them.
And this is not the limit!
Video
In our video you will find interesting examples of solving problems of finding speed, time and distance.
For all stages of the gearbox and additional gearbox, vehicle speed values are calculated depending on the engine speed (by agreement with the manager, calculations can be made only for the highest stage of the additional gearbox).
The calculation is carried out according to the formula
Where v - vehicle speed, km/h;
n - engine crankshaft rotation speed, rpm;
rTO - rolling radius, m;
And 0 - final drive gear ratio;
AndTo - gear ratio of the calculated gearbox stage;
Andd - gear ratio of the calculated stage of the additional (transfer) box.
The crankshaft rotation speed values are taken to be the same as when constructing the external speed characteristic.
Calculated values vt are entered in column 4 of the table. 2.1. Graphs of the dependence of the vehicle speed on the engine crankshaft rotation speed are a series of rays emerging at different angles from the origin of coordinates, Figure 2.2.
Rice. 2.2 Dependence of vehicle speed on crankshaft rotation speed in gears.
2.6. Traction characteristics and traction balance of the vehicle
The traction characteristic is the dependence of the vehicle's traction force on the speed of movement through the gears. Traction values RT are calculated at individual points using the formula
Where MTO - engine torque, Nm;
η T - Transmission efficiency.
Calculation results RT are entered in column 7 of the table. 2.1, and dependency graphs are built from them RT = f(V) by transmission.
The traction balance of a car is described by the traction or force balance equation
RT = Rd+ RV+ RAnd, (2.27)
Where RT - vehicle traction force, N;
Rd - total road resistance force, N;
RV - air resistance force, N;
RAnd - inertia force of the car, N.
Magnitude Rd determined by the expression
Rd = Gaψ , (2.28)
Where Ga - total vehicle weight, N; ψ - total road resistance coefficient.
The total drag coefficient of the road is a value that depends on the speed of the vehicle. However, taking this dependence into account greatly complicates the performance of traction calculations and at the same time does not provide clarification that is important for practice. Therefore, when performing traction calculations, it is recommended to take the value ψ constant, equal to the value that was calculated for the maximum vehicle speed when determining the engine power required to drive at maximum speed, i.e. accept everywhere ψ=ψ v.
For any one selected value ψ magnitude Rd remains constant for all calculated points in all gears. Therefore the value Rd is counted once and is not entered into the table. On the traction characteristic graph, the dependence PT= f(v) is represented as a straight line parallel to the abscissa axis.
Rice. 2.3 Traction characteristics of the car.
Air resistance force RV amounts to
Where WithX - longitudinal aerodynamic force coefficient;
RV - air density, kg/m3;
ToV - streamlining coefficient, kg/m 3 ;
F - frontal area of the car, m;
vV - air flow speed relative to the car, km/h.
When calculating, you can specify ρ V=1.225 kg/m. The speed of the air flow is usually assumed to be equal to the speed of the vehicle.
Values RV are calculated for all points and entered in column 5 of the table. 2.1. Dependency graph RV of velocity is a parabola passing through the origin.
For the convenience of further analysis, this graph is shifted upward by an amount equal toR d (on the scale accepted for forces). In fact, with this construction, this graph expresses the dependence( P V + P d )= f ( v ).
Vehicle inertia force RAnd after calculation Rd And RV can be defined as the closing term of the force balance
(2.30)
On the graph the valueR And determined by a straight line segment drawn for the desired speed value parallel to the ordinate axis, between the points of intersection of this straight line of the graphs P T = f [ v ) And( P d + P V )= f ( v ). If a given speed can be achieved in several gears, then each of these gears will have its own inertia force value. Calculated values R And should be entered in column 6 of the table. 2.1.
The RT value is entered in column 7 of the table. 2.1. The traction characteristics of the car are shown in Fig. 2.3.
Definition
Instant speed(or more often just the speed) of a material point is a physical quantity equal to the first derivative of the radius vector of the point with respect to time (t). Speed is usually denoted by the letter v.
This is a vector quantity. Mathematically, the definition of the instantaneous velocity vector is written as:
The velocity has a direction indicating the direction of movement of the material point and lies on the tangent to the trajectory of its movement. The velocity module can be defined as the first derivative of the path length (s) with respect to time:
Velocity characterizes the speed of movement in the direction of motion of a point in relation to the coordinate system under consideration.
Speed in different coordinate systems
Projections of velocity on the axes of the Cartesian coordinate system will be written as:
Therefore, the velocity vector in Cartesian coordinates can be represented:
where are the unit unit vectors. In this case, the magnitude of the velocity vector is found using the formula:
In cylindrical coordinates, the velocity module is calculated using the formula:
in a spherical coordinate system:
Special cases of formulas for calculating speed
If the velocity module does not change over time, then such motion is called uniform (v=const). With uniform motion, speed can be calculated using the formula:
where s is the length of the path, t is the time during which the material point covered the path s.
With accelerated motion, the speed can be found as:
where is the acceleration of the point, is the period of time during which the speed is considered.
If the movement is uniformly variable, then the following formula is used to calculate the speed:
where is the initial speed of movement, .
Speed units
The basic unit of measurement of speed in the SI system is: [v] = m/s 2
In GHS: [v]=cm/s 2
Examples of problem solving
Example Exercise.
The motion of material point A is given by the equation: . The point began its movement at t 0 =0 s. How will the point in question move relative to the X axis at time t = 0.5 s. Solution.
Let's find an equation that will specify the speed of the material point under consideration; for this, from the function x=x(t), which is specified in the conditions of the problem, we take the first derivative with respect to time, we get:
To determine the direction of movement, we substitute the time specified in the condition into the function we obtained for the speed v=v(t) in (1.1) and compare the result with zero:
Since we obtained that the speed at the indicated moment of time is negative, therefore, the material point moves against the X axis. Answer.
Examples of problem solving
Example Against the X axis.
where speed is in m/s, time is in s. What is the coordinate of the point at a time equal to 10 s; at what point in time will the point be at a distance of 10 m from the origin? Consider that at t=0 c the point of origin moves from the origin along the X axis.
The motion of material point A is given by the equation: . The point began its movement at t 0 =0 s. How will the point in question move relative to the X axis at time t = 0.5 s. The point moves along the X axis, the relationship between the x coordinate and the speed of movement is determined by the formula.
How to solve motion problems? Formula for the relationship between speed, time and distance. Problems and solutions.
Formula for the dependence of time, speed and distance for grade 4: how is speed, time, distance indicated?
People, animals or cars can move at a certain speed. In a certain time they can travel a certain distance. For example: today you can walk to your school in half an hour. You walk at a certain speed and cover 1000 meters in 30 minutes. The path that is overcome is denoted in mathematics by the letter S. Speed is indicated by the letter v. And the time it takes to travel is indicated by the letter t.
- Path - S
- Speed - v
- Time - t
If you are late for school, you can cover the same route in 20 minutes by increasing your speed. This means that the same path can be covered in different times and at different speeds.
How does travel time depend on speed?
The higher the speed, the faster the distance will be covered. And the lower the speed, the more time it will take to complete the journey.
How to find time knowing speed and distance?
In order to find the time it took to travel a path, you need to know the distance and speed. If you divide the distance by the speed, you get the time. An example of such a task:
Problem about the Hare. The Hare ran away from the Wolf at a speed of 1 kilometer per minute. He ran 3 kilometers to his hole. How long did it take the Hare to reach the hole?
How can you easily solve motion problems where you need to find distance, time or speed?
- Read the problem carefully and determine what is known from the problem statement.
- Write this information on your draft.
- Also write what is unknown and what needs to be found
- Use the formula for problems about distance, time and speed
- Enter known data into the formula and solve the problem
Solution for the problem about the Hare and the Wolf.
- From the conditions of the problem we determine that we know the speed and distance.
- We also determine from the conditions of the problem that we need to find the time it took for the hare to run to the hole.
We write this data in the draft, for example like this:
Time - unknown
Now let’s write the same thing in mathematical symbols:
S - 3 kilometers
V - 1 km/min
t — ?
We remember and write down in a notebook the formula for finding time:
t=S:v
t = 3: 1 = 3 minutes
How to find speed if time and distance are known?
In order to find the speed, if time and distance are known, you need to divide the distance by time. An example of such a task:
The Hare ran away from the Wolf and ran 3 kilometers to its hole. He covered this distance in 3 minutes. How fast did the Hare run?
Solution to the motion problem:
- We write down in the draft that we know the distance and time.
- From the conditions of the problem we determine that we need to find the speed
- Let's remember the formula for finding speed.
Formulas for solving such problems are shown in the picture below.
Formulas for solving problems about distance, time and speed We substitute the known data and solve the problem:
Distance to the hole - 3 kilometers
The time it took the Hare to reach the hole - 3 minutes
Speed - unknown
Let's write these known data in mathematical symbols
S - 3 kilometers
t - 3 minutes
v — ?
We write down the formula for finding speed
v=S:t
Now let’s write down the solution to the problem in numbers:
v = 3: 3 = 1 km/min
How to find distance if time and speed are known?
To find the distance, if the time and speed are known, you need to multiply the time by the speed. An example of such a task:
The Hare ran away from the Wolf at a speed of 1 kilometer in 1 minute. It took him three minutes to reach the hole. How far did the Hare run?
Solution to the problem: We write down in the draft what we know from the problem statement:
The speed of the Hare is 1 kilometer in 1 minute
The time the Hare ran to the hole was 3 minutes
Distance - unknown
Now, let's write the same thing in mathematical symbols:
v — 1 km/min
t - 3 minutes
S — ?
Let us recall the formula for finding the distance:
S = v ⋅ t
Now let’s write down the solution to the problem in numbers:
S = 3 ⋅ 1 = 3 km
How to learn to solve more complex problems?
To learn how to solve more complex problems, you need to understand how simple ones are solved, remember what signs indicate distance, speed and time. If you can’t remember mathematical formulas, you need to write them down on a piece of paper and always keep them at hand while solving problems. Solve simple problems with your child that you can come up with on the go, for example, while walking.
A child who can solve problems can be proud of himself
When solving problems about speed, time and distance, they often make a mistake because they forgot to convert units of measurement.
IMPORTANT: The units of measurement can be any, but if the same problem has different units of measurement, convert them to the same ones. For example, if the speed is measured in kilometers per minute, then the distance must be presented in kilometers and the time in minutes.
For the curious: The now generally accepted system of measures is called metric, but this was not always the case, and in the old days other units of measurement were used in Rus'.
Problem about a boa constrictor: The baby elephant and the monkey measured the length of the boa constrictor in steps. They moved towards each other. The speed of the monkey was 60 cm in one second, and the speed of the baby elephant was 20 cm in one second. They took 5 seconds to measure. What is the length of a boa constrictor? (solution under the picture)
Solution:
From the conditions of the problem we determine that we know the speed of the monkey and the baby elephant and the time it took them to measure the length of the boa constrictor.
Let's write down this data:
Monkey speed - 60 cm/sec
Baby elephant speed - 20 cm/sec
Time - 5 seconds
Distance unknown
Let's write this data in mathematical symbols:
v1 — 60 cm/sec
v2 — 20 cm/sec
t - 5 seconds
S — ?
Let's write the formula for distance if the speed and time are known:
S = v ⋅ t
Let's calculate how far the monkey has traveled:
S1 = 60 ⋅ 5 = 300 cm
Now let’s calculate how far the baby elephant has walked:
S2 = 20 ⋅ 5 = 100 cm
Let's sum up the distance the monkey walked and the distance the baby elephant walked:
S = S1 + S2 = 300 + 100 = 400 cm
Graph of body speed versus time: photo
The distance covered at different speeds is covered in different times. The higher the speed, the less time it will take to move.
Table 4 class: speed, time, distance
The table below shows data for which you need to come up with problems and then solve them.
| № | Speed (km/h) | Time (hour) | Distance (km) |
| 1 | 5 | 2 | ? |
| 2 | 12 | ? | 12 |
| 3 | 60 | 4 | ? |
| 4 | ? | 3 | 300 |
| 5 | 220 | ? | 440 |
You can use your imagination and come up with problems for the table yourself. Below are our options for the task conditions:
- Mom sent Little Red Riding Hood to her grandmother. The girl was constantly distracted and walked through the forest slowly, at a speed of 5 km/hour. She spent 2 hours on the way. How far did Little Red Riding Hood travel during this time?
- Postman Pechkin was carrying a parcel on a bicycle at a speed of 12 km/h. He knows that the distance between his house and Uncle Fedor's house is 12 km. Help Pechkin calculate how long it will take to travel?
- Ksyusha's dad bought a car and decided to take his family to the sea. The car was traveling at a speed of 60 km/h and the journey took 4 hours. What is the distance between Ksyusha’s house and the sea coast?
- The ducks gathered in a wedge and flew to warmer climes. The birds flapping their wings tirelessly for 3 hours and covered 300 km during this time. What was the speed of the birds?
- The AN-2 plane flies at a speed of 220 km/h. He took off from Moscow and flies to Nizhny Novgorod, the distance between these two cities is 440 km. How long will the plane travel?
Answers to the given problems can be found in the table below:
| № | Speed (km/h) | Time (hour) | Distance (km) |
| 1 | 5 | 2 | 10 |
| 2 | 12 | 1 | 12 |
| 3 | 60 | 4 | 240 |
| 4 | 100 | 3 | 300 |
| 5 | 220 | 2 | 440 |
Examples of solving problems on speed, time, distance for grade 4
If there are several objects of movement in one task, you need to teach the child to consider the movement of these objects separately and only then together. An example of such a task:
Two friends Vadik and Tema decided to take a walk and left their houses towards each other. Vadik was riding a bicycle, and Tema was walking. Vadik was driving at a speed of 10 km/h, and Tema was walking at a speed of 5 km/h. An hour later they met. What is the distance between Vadik's and Tema's houses?
This problem can be solved using the formula for the dependence of distance on speed and time.
S = v ⋅ t
The distance that Vadik traveled on a bicycle will be equal to his speed multiplied by the travel time.
S = 10 ⋅ 1 = 10 kilometers
The distance traveled by Theme is calculated similarly:
S = v ⋅ t
We substitute the digital values of its speed and time into the formula
S = 5 ⋅ 1 = 5 kilometers
The distance that Vadik traveled must be added to the distance that Tema traveled.
10 + 5 = 15 kilometers
How to learn to solve complex problems that require logical thinking?
To develop a child’s logical thinking, you need to solve simple and then complex logical problems with him. These tasks may consist of several stages. You can move from one stage to another only if the previous one has been solved. An example of such a task:
Anton was riding a bicycle at a speed of 12 km/h, and Lisa was riding a scooter at a speed 2 times less than Anton's, and Denis was walking at a speed 2 times less than Lisa's. What is Denis's speed?
To solve this problem, you must first find out Lisa’s speed and only then Denis’s speed.
Who goes faster? Problem about friends Sometimes textbooks for grade 4 contain difficult problems. An example of such a task:
Two cyclists rode from different cities towards each other. One of them was in a hurry and rushing at a speed of 12 km/h, and the second was driving slowly at a speed of 8 km/h. The distance between the cities from which the cyclists left is 60 km. How far will each cyclist travel before they meet? (solution under photo)
Solution:
- 12+8 = 20 (km/h) is the total speed of two cyclists, or the speed at which they approached each other
- 60 : 20 = 3 (hours) - this is the time after which the cyclists met
- 3 ⋅ 8 = 24 (km) is the distance traveled by the first cyclist
- 12 ⋅ 3 = 36 (km) is the distance traveled by the second cyclist
- Check: 36+24=60 (km) is the distance traveled by two cyclists.
- Answer: 24 km, 36 km.
Encourage children to solve such problems in the form of a game. They may want to create their own problem about friends, animals or birds.
VIDEO: Movement problems
To find speed, time and distance, you need to take a school textbook and read it)) I liked such problems.
Speed is measured by the distance traveled in a certain time, so we divide the distance by time and get, for example, kilometers per hour. Well, the remaining quantities can be calculated based on this formula.
This question applies to junior high school math.
The distance can be found by multiplying the speed and time taken to cover this distance.
And accordingly, time is equal to distance divided by speed.
- To find out the speed, divide the distance by time;
- To find out the time, divide the distance by the speed;
- To find out the distance, we multiply the speed by the time.
Everything is quite simple and easy, since everyone at school knew this formula - you just need to remember!)
Some people remember faster when they read and look, so by looking at these formulas proposed in the image, you can remember them almost for the rest of your life.
All three formulas are interconnected and one follows the other.
Movement problems are one of the important topics for students. To solve problems, you need to know the rules for finding quantities. To find the distance, you need to multiply the speed by the time; to find the time, you need to divide the distance by the speed. To find the speed, you need to divide the distance by the time.
If the body moves uniformly, i.e. at a constant speed, it is very easy to determine one of these quantities if the other two are known.
Speed, distance and time are denoted by the letters V, S, t, respectively.
Speed: V = S/t
Distance: S = V*t
Time: t = S/V
To find the distance, you need to multiply the speed by the travel time.
To find the speed, you need to divide the distance by the time.
In order to find travel time, you need to divide the distance by the speed.
Well, here’s a picture to go with it all, here there are formulas with all the designations.
To find physical quantities such as speed (V), time (t) and distance (S), you need to know that these quantities depend on movement.
Movement can be equally accelerated, equally slow, or uniform.
With equal acceleration and equal deceleration, the speed depends on time. And with uniform speed, the speed does not change, i.e. is constant.
The formulas are presented below:
Speed, time, distance - all these are physical quantities that are somehow related to movement. Movement can be either uniform or uniformly accelerated (as well as uniformly slow). While in uniform motion the body moves at a constant speed, which does not depend on time, the uniformly accelerated speed can change over time.
How to find one of the three speed values if we know the other two?
Well, to find out the time you need to divide the distance by the speed; of course, the values of the distance and speed must be known. To find out the speed, you need to divide the distance by time, for example, you get a common value - mph.
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