How to determine the speed of movement. How to find average 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).
Speed is a quantity that describes how quickly an object moves from point A to point B. It is denoted by the Latin letter V - short for the Latin velocitas - speed. Speed can be found if you know the time (t) during which the object moved and the distance (S) that the object traveled.
To calculate speed, use the path formula: V=S/t. For example, in 12 seconds the object moved 60 meters, which means its speed was 5 m/s (V=60/12=5). Use the same units when comparing the speed of two different objects. The basic unit of speed in the International System of Units is meters per second, or m/s for short. Also common are kilometers per hour, kilometers per second, meters per minute, and meters per second. In English-speaking countries, miles per second, miles per hour, feet per second and feet per minute are used. Remember, the accuracy of speed determination depends on the nature of the movement. Most accurately, the path formula helps to find the speed of uniform motion - an object covers the same distance in equal periods of time. However, uniform motion is very rare in the real world. This is, for example, the movement of the second hand on a watch or the rotation of the Earth around the Sun. In case of uneven movement, for example, walking around the city, the path formula helps to find the average speed.
Let's turn a school physics lesson into an exciting game! In this article, our heroine will be the formula “Speed, time, distance”. Let's look at each parameter separately and give interesting examples.
Speed
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 drove for some time, let's say 5 km/h. That is, in 1 hour he walked 5 kilometers.
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.
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 modulus 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 path length, 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 set 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. Against the X axis.
Example
Exercise. The velocity of a material point is a function of time of the form:
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.
Solution. The point moves along the X axis, the relationship between the x coordinate and the speed of movement is determined by the formula.
The concept of time (as well as distance and speed) is a physical quantity. It characterizes the interval during which an object changes its properties and is used in physics and mathematics to solve problems involving motion.
As an example, let's try to find time if the distance and speed are known, and also consider reverse methods for calculating unknown quantities.
Quick navigation through the article
Determining the time
To determine time, they usually use the common formula: t=S/v, where t is time, S is distance, and v is speed.
Thus, using simple mathematical operations, you can calculate any of these quantities, knowing the other two. In this case, we have speed and distance values. To find out the time, we divide the distance by the speed.
The same formula will help calculate the speed provided that the distance and time are known. To do this, we perform simple mathematical operations with ordinary fractions.
Determining the speed
From the formula by which we calculated the time, we calculate the speed. This is a value equal to the distance traveled per unit time.
To find the speed value, you need to place it on one side of the equal sign and the other values on the other. To calculate the denominator in this equation, you need to divide the numerator by the value on the other side of the equal sign. That is, we divide the distance by time and get the following formula: v=S/t
Determining the distance
By analogy, we calculate the distance. It will be determined by the product of time and speed: S=v*t