Uniform movement is movement at a constant speed. That is, in other words, the body must travel the same distance in equal periods of time. For example, if a car covers a distance of 50 kilometers for every hour of its journey, then such movement will be uniform.
Generally, uniform motion is very rarely encountered in real life. Examples of uniform motion in nature include the rotation of the Earth around the Sun. Or, for example, the end of the second hand of a watch will also move evenly.
Calculation of speed during uniform motion
The speed of a body during uniform motion will be calculated using the following formula.
- Speed = path / time.
If we denote the speed of movement by the letter V, the time of movement by the letter t, and the path traveled by the body by the letter S, we obtain the following formula.
- V=s/t.
The speed unit is 1 m/s. That is, a body travels a distance of one meter in a time equal to one second.
Movement with variable speed is called uneven movement. Most often, all bodies in nature move unevenly. For example, when a person walks somewhere, he moves unevenly, that is, his speed will change throughout the entire journey.
Calculation of speed during uneven movement
With uneven movement, the speed changes all the time, and in this case we talk about the average speed of movement.
The average speed of uneven movement is calculated by the formula
- Vcp=S/t.
From the formula for determining speed, we can obtain other formulas, for example, to calculate the distance traveled or the time that the body moved.
Calculation of path for uniform motion
To determine the path traveled by a body during uniform motion, it is necessary to multiply the speed of movement of the body by the time that this body moved.
- S=V*t.
That is, knowing the speed and time of movement, we can always find the path.
Now, we get a formula for calculating the time of movement, given the known speed of movement and the distance traveled.
Calculation of time during uniform motion
In order to determine the time of uniform motion, it is necessary to divide the distance traveled by the body by the speed with which this body moved.
- t=S/V.
The formulas obtained above will be valid if the body performed uniform motion.
When calculating the average speed of uneven movement, it is assumed that the movement was uniform. Based on this, to calculate the average speed of uneven movement, the distance or time of movement, the same formulas are used as for uniform movement.
Path calculation for uneven movement
We find that the path traveled by a body during uneven motion is equal to the product of the average speed and the time the body moved.
- S=Vcp*t
Calculation of time for uneven movement
The time required to travel a certain path during uneven movement is equal to the quotient of the path divided by the average speed of uneven movement.
- t=S/Vcp.
The graph of uniform motion in coordinates S(t) will be a straight line.
What is “speed”? You can watch how one car goes faster, another goes slower; one person walks at a brisk pace, the other takes his time. Cyclists also travel at different speeds. Yes! Precisely speed. What does it mean? Of course, the distance that a person has walked. the car passed in a certain amount of time. Let's assume that a person's speed is 5 km/h. That is, in 1 hour he walked 5 kilometers.
How to find speed, time, distance? Let's start with speed. Look carefully, what is it measured in? Naturally, km/h, m/s. There are other units of measurement, for example, km/s (in astronautics), mm/h (in biochemistry). Pay attention to what comes before and after the “/” sign. Firstly, it means “fraction”, which means that the numerator is mm, km, m, and the denominator is h, s, min. Secondly, it seems like a formula, doesn't it? Kilometers, meters - distance, length, and hour, second, minute - time. Here's a hint for you. To make it easier to remember how to find speed, look at the units (km/h, m/s). In one word:
Time
What is time? Of course, it depends on the speed. For example, you are waiting at the doorstep of the house for your mother and older brother. They are coming from the store. My brother got there much earlier. Mom had to wait another 5 minutes. Why? Because they walked at different speeds. Of course, in order to get to your destination faster, you need to increase your speed: speed up your pace, press the gas in the car harder, accelerate on a bicycle. Just be careful and vigilant when in a hurry so as not to crash into someone or something. 
How to find time? Speed has a clue - km/h. What about time? Firstly, time is measured in minutes, seconds, hours. The formula “speed, time, distance” is transformed here as follows:
time t[sec., min., h]=S[m, mm, km]/v[m/s, mm/min, km/h].
If you transform the fraction according to all the rules of mathematics, reducing the distance (length) parameter, then only a second, a minute or an hour will remain.
Distance, length of path traveled
It will be easier to navigate here, most likely, for motorists who have a mileage meter in their car. They will be able to determine how many kilometers they have traveled, and they also know the speed. But since the movement is uneven, it will not be possible to establish the exact time of movement if we only take the average speed. 
The formula for path (distance) is the product of speed and time. Of course, the most convenient and accessible parameter is time. Everyone has a watch. The pedestrian speed is not strictly 5 km/h, but approximately. Therefore, there may be an error here. In this case, you better take a map of the area. Notice the scale. It should indicate how many kilometers or meters are in 1 cm. Attach a ruler and measure the length. For example, there is a direct road from home to a music school. The segment turned out to be 5 cm. And the scale indicates 1 cm = 200 m. This means that the real distance is 200 * 5 = 1000 m = 1 km. How long does it take you to cover this distance? In half an hour? In technical terms, 30 minutes = 0.5 hours = (1/2) hours. If we solve the problem, it turns out that you are walking at a speed of 2 km/h. The formula “speed, time, distance” will always help you solve the problem.
Don't miss out!
I advise you not to miss very important points. When you are given a task, look carefully at what units of measurement the parameters are given in. The author of the task can cheat. Will write in given:
A man rode a bicycle along the sidewalk 2 kilometers in 15 minutes. Do not rush to immediately solve the problem using the formula, otherwise you will end up with nonsense, and the teacher will not count it for you. Remember that under no circumstances should you do this: 2 km/15 min. Your unit of measurement will be km/min, not km/h. You need to achieve the latter. Convert minutes to hours. How to do it? 15 minutes is 1/4 hour or 0.25 hours. Now you can safely 2km/0.25h=8 km/h. Now the problem has been solved correctly.
This is how easy it is to remember the formula “speed, time, distance.” Just follow all the rules of mathematics and pay attention to the units of measurement in the problem. If there are nuances, as in the example discussed just above, immediately convert to the SI system of units, as expected.
How to find speed, time, distance?
- If time and speed are known, and it is necessary to find the distance during uniform motion, then we apply the following formula:
- When you need to calculate the speed for already known values of distance and time when moving at a uniform speed, we calculate as follows:
- If you want to determine time while the distance as well as the speed are known, provided that the movement is uniform:
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, 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 the 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.
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 is the speed of the object,
- 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:
v=100 km/1.5 h=66.66 km/h
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 velocities on individual sections of the path S, n is the number of these sections, vav is the average speed of the object along the entire path.
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 is the 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, path 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!
Speed is a function of time and is determined by both absolute value and direction. Often in physics problems it is required to find the initial speed (its magnitude and direction) that the object under study had at the zero moment of time. Various equations can be used to calculate initial velocity. Based on the data given in the problem statement, you can choose the most appropriate formula that will easily obtain the desired answer.
Steps
Finding the initial speed from the final speed, acceleration and time
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When solving a physics problem, you need to know what formula you will need. To do this, the first step is to write down all the data given in the problem statement. If the final speed, acceleration and time are known, it is convenient to use the following relationship to determine the initial speed:
- V i = V f - (a * t)
- V i- starting speed
- V f- final speed
- a- acceleration
- t- time
- Please note that this is the standard formula used to calculate initial velocity.
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Having written out all the initial data and written down the necessary equation, you can substitute known quantities into it. It is important to carefully study the problem statement and carefully write down each step when solving it.
- If you made a mistake anywhere, you can easily find it by looking through your notes.
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Solve the equation. Substituting known values into the formula, use standard transformations to obtain the desired result. If possible, use a calculator to reduce the likelihood of miscalculations.
- Suppose an object, moving east at an acceleration of 10 meters per second squared for 12 seconds, accelerates to a final speed of 200 meters per second. It is necessary to find the initial speed of the object.
- Let's write down the initial data:
- V i = ?, V f= 200 m/s, a= 10 m/s 2, t= 12 s
- Let's multiply the acceleration by time: a*t = 10 * 12 =120
- Subtract the resulting value from the final speed: V i = V f – (a * t) = 200 – 120 = 80 V i= 80 m/s to the east
- m/s
- Suppose an object, moving east at an acceleration of 10 meters per second squared for 12 seconds, accelerates to a final speed of 200 meters per second. It is necessary to find the initial speed of the object.
Finding the initial speed from the distance traveled, time and acceleration
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Use the appropriate formula. When solving any physical problem, it is necessary to choose the appropriate equation. To do this, the first step is to write down all the data given in the problem statement. If the distance traveled, time and acceleration are known, the following relationship can be used to determine the initial speed:
- This formula includes the following quantities:
- V i- starting speed
- d- distance traveled
- a- acceleration
- t- time
- This formula includes the following quantities:
-
Substitute known quantities into the formula.
- If you make a mistake in a decision, you can easily find it by looking through your notes.
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Solve the equation. Substitute known values into the formula and use standard transformations to find the answer. If possible, use a calculator to reduce the chance of miscalculation.
- Let's say an object moves in a westward direction with an acceleration of 7 meters per second squared for 30 seconds, traveling 150 meters. It is necessary to calculate its initial speed.
- Let's write down the initial data:
- V i = ?, d= 150 m, a= 7 m/s 2, t= 30 s
- Let's multiply the acceleration by time: a*t = 7 * 30 = 210
- Let's divide the product into two: (a * t) / 2 = 210 / 2 = 105
- Let's divide the distance by time: d/t = 150 / 30 = 5
- Subtract the first quantity from the second: V i = (d / t) - [(a * t) / 2] = 5 – 105 = -100 V i= -100 m/s westward
- Write the answer in the correct form. It is necessary to specify the units of measurement, in our case meters per second, or m/s, as well as the direction of movement of the object. If you do not specify a direction, the answer will be incomplete, containing only the value of the speed without information about which direction the object is moving.
- Let's say an object moves in a westward direction with an acceleration of 7 meters per second squared for 30 seconds, traveling 150 meters. It is necessary to calculate its initial speed.
Finding the initial speed from the final speed, acceleration and distance traveled
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Use the appropriate equation. To solve a physical problem, you need to choose the appropriate formula. The first step is to write down all the initial data specified in the problem statement. If the final speed, acceleration and distance traveled are known, it is convenient to use the following relationship to determine the initial speed:
- V i = √
- This formula contains the following quantities:
- V i- starting speed
- V f- final speed
- a- acceleration
- d- distance traveled
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Substitute known quantities into the formula. After you have written down all the initial data and written down the necessary equation, you can substitute known quantities into it. It is important to carefully study the problem statement and carefully write down each step when solving it.
- If you make a mistake somewhere, you can easily find it by reviewing the progress of the solution.
-
Solve the equation. Substituting known values into the formula, use the necessary transformations to obtain the answer. If possible, use a calculator to reduce the likelihood of miscalculations.
- Suppose an object moves in a northerly direction with an acceleration of 5 meters per second squared and, after traveling 10 meters, has a final speed of 12 meters per second. It is necessary to find its initial speed.
- Let's write down the initial data:
- V i = ?, V f= 12 m/s, a= 5 m/s 2, d= 10 m
- Let's square the final speed: V f 2= 12 2 = 144
- Multiply the acceleration by the distance traveled and by 2: 2*a*d = 2 * 5 * 10 = 100
- Subtract the result of the multiplication from the square of the final speed: V f 2 - (2 * a * d) = 144 – 100 = 44
- Let's take the square root of the resulting value: = √ = √44 = 6,633 V i= 6.633 m/s northward
- Write the answer in the correct form. The units of measurement must be specified, i.e. meters per second, or m/s, as well as the direction of movement of the object. If you do not specify a direction, the answer will be incomplete, containing only the value of the speed without information about which direction the object is moving.
- Suppose an object moves in a northerly direction with an acceleration of 5 meters per second squared and, after traveling 10 meters, has a final speed of 12 meters per second. It is necessary to find its initial speed.
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 (initial) – the initial speed of the object,
- a – acceleration of the body,
- t – total travel time.
How to find speed - uneven motion
In this case, there is a situation where the body passed 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 complete 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.
- Given a known linear speed of an object for each point on the path of motion 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 of the 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 departure.
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).
In rectilinear uniformly accelerated motion the body
- moves along a conventional straight line,
- its speed gradually increases or decreases,
- over equal periods of time, the speed changes by an equal amount.
For example, a car starts moving from a state of rest along a straight road, and up to a speed of, say, 72 km/h it moves uniformly accelerated. When the set speed is reached, the car moves without changing speed, i.e. uniformly. With uniformly accelerated motion, its speed increased from 0 to 72 km/h. And let the speed increase by 3.6 km/h for every second of movement. Then the time of uniformly accelerated movement of the car will be equal to 20 seconds. Since acceleration in SI is measured in meters per second squared, acceleration of 3.6 km/h per second must be converted into the appropriate units. It will be equal to (3.6 * 1000 m) / (3600 s * 1 s) = 1 m/s 2.
Let's say that after some time of driving at a constant speed, the car began to slow down to stop. The movement during braking was also uniformly accelerated (over equal periods of time, the speed decreased by the same amount). In this case, the acceleration vector will be opposite to the velocity vector. We can say that the acceleration is negative.
So, if the initial speed of a body is zero, then its speed after a time of t seconds will be equal to the product of acceleration and this time:
When a body falls, the acceleration of gravity “works”, and the speed of the body at the very surface of the earth will be determined by the formula:
If the current speed of the body and the time it took to develop such a speed from a state of rest are known, then the acceleration (i.e. how quickly the speed changed) can be determined by dividing the speed by the time:
However, the body could begin uniformly accelerated motion not from a state of rest, but already possessing some speed (or it was given an initial speed). Let's say you throw a stone vertically down from a tower using force. Such a body is subject to a gravitational acceleration equal to 9.8 m/s 2 . However, your strength gave the stone even more speed. Thus, the final speed (at the moment of touching the ground) will be the sum of the speed developed as a result of acceleration and the initial speed. Thus, the final speed will be found according to the formula:
However, if the stone was thrown upward. Then its initial speed is directed upward, and the acceleration of free fall is directed downward. That is, the velocity vectors are directed in opposite directions. In this case (as well as during braking), the product of acceleration and time must be subtracted from the initial speed:
From these formulas we obtain the acceleration formulas. In case of acceleration:
at = v – v 0
a = (v – v 0)/t
In case of braking:
at = v 0 – v
a = (v 0 – v)/t
In the case when a body stops with uniform acceleration, then at the moment of stopping its speed is 0. Then the formula is reduced to this form:
Knowing the initial speed of the body and the braking acceleration, the time after which the body will stop is determined:
Now let's print formulas for the path that a body travels during rectilinear uniformly accelerated motion. The graph of speed versus time for rectilinear uniform motion is a segment parallel to the time axis (usually the x axis is taken). The path is calculated as the area of the rectangle under the segment. That is, by multiplying speed by time (s = vt). With rectilinear uniformly accelerated motion, the graph is a straight line, but not parallel to the time axis. This straight line either increases in the case of acceleration or decreases in the case of braking. However, path is also defined as the area of the figure under the graph.
In rectilinear uniformly accelerated motion, this figure is a trapezoid. Its bases are a segment on the y-axis (speed) and a segment connecting the end point of the graph with its projection on the x-axis. The sides are the graph of speed versus time itself and its projection onto the x-axis (time axis). The projection onto the x-axis is not only the side side, but also the height of the trapezoid, since it is perpendicular to its bases.
As you know, the area of a trapezoid is equal to half the sum of the bases and the height. The length of the first base is equal to the initial speed (v 0), the length of the second base is equal to the final speed (v), the height is equal to time. Thus we get:
s = ½ * (v 0 + v) * t
Above was given the formula for the dependence of the final speed on the initial and acceleration (v = v 0 + at). Therefore, in the path formula we can replace v:
s = ½ * (v 0 + v 0 + at) * t = ½ * (2v 0 + at) * t = ½ * t * 2v 0 + ½ * t * at = v 0 t + 1/2at 2
So, the distance traveled is determined by the formula:
s = v 0 t + at 2 /2
(This formula can be arrived at by considering not the area of the trapezoid, but by summing up the areas of the rectangle and right triangle into which the trapezoid is divided.)
If the body begins to move uniformly accelerated from a state of rest (v 0 = 0), then the path formula simplifies to s = at 2 /2.
If the acceleration vector was opposite to the speed, then the product at 2 /2 must be subtracted. It is clear that in this case the difference between v 0 t and at 2 /2 should not become negative. When it becomes zero, the body will stop. A braking path will be found. Above was the formula for the time until a complete stop (t = v 0 /a). If we substitute the value t into the path formula, then the braking path is reduced to the following formula.