Speed is a physical quantity that measures the change in the position of an object based on time, that is, how quickly it is moving in a given instant of time. If you have ever had the opportunity to observe the speedometer of a car while it is in motion, you were witnessing the instant measurement of the speed of the vehicle: the more the pointer moves towards the full scale, the faster the vehicle travels. There are several ways to calculate the speed which depend on the type of information we have available. Normally use the equation Speed = Space / Time (or more simply v = s / t) is the simplest way to calculate the speed of an object.
Steps
Part 1 of 3: Using the Standard Equation for Speed Calculation
Step 1. Identify the distance that the object covered during the movement it made
The basic equation that most people use to calculate the speed of a vehicle or object is very simple to solve. The first thing to know is the distance traveled by the object under examination. In other words, the distance that separates the starting point from the arrival point.
It is much easier to understand the meaning of this equation with an example. Let's say we are sitting in the car heading to a theme park that is far away 160 km from the starting point. The next steps show how to use this information to solve the equation.
Step 2. Determine the time the object under examination took to cover the entire distance
The next data that you need to know in order to solve the problem is the time taken by the object to complete the entire path. In other words, how much time did it take to move from the starting point to the arrival point.
In our example we assume that we have reached the theme park in two hours travel exact.
Step 3. To get the speed of the object under examination, we divide the space it traveled by the time it took
To calculate the speed of any object it is necessary to have only these two simple information. The relationship between the distance traveled and the time taken will give us as a result the speed of the observed object.
In our example we will get 160 km / 2 hours = 80 km / h.
Step 4. Don't forget to add the units of measure
A very important step to be able to correctly express the results obtained is to use the units of measurement in the right way (for example, kilometers per hour, miles per hour, meters per second, etc.). Reporting the result of the calculations without adding any unit of measurement would make it impossible for those who have to interpret it or simply read it to be able to understand its meaning. In addition, in the case of a test or a school test you would risk getting a lower grade.
The speed unit is represented the ratio between the unit of measurement of the distance traveled and that of the time taken. Since in our example we measured space n kilometers and time in hours, the correct unit to use is i km / h, that is, kilometers per hour.
Part 2 of 3: Solving Intermediate Problems
Step 1. Use the inverse equation to calculate space or time
After understanding the meaning of the equation for calculating the speed of an object, it can be used to calculate all the quantities under consideration. For example, assuming we know the speed of an object and one of the other two variables (distance or time), we can modify the starting equation to be able to trace the missing data.
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Let's assume we know that a train has traveled at a speed of 20 km / h for 4 hours and we need to calculate the distance it has managed to travel. In this case we have to modify the basic equation for calculating the speed as follows:
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- Speed = Space / Time;
- Speed × Time = (Space / Time) × Time;
- Speed × Time = Space;
- 20 km / h × 4 h = Space = 80 km.
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Step 2. Convert the units of measurement as needed
Sometimes it may be necessary to report the speed using a different unit of measurement than the one obtained through the calculations. In this case, a conversion factor must be used in order to express the result obtained with the correct unit of measurement. To perform the conversion it is sufficient to simply express the relationship between the units of measurement in question in the form of a fraction or multiplication. When converting, you must use a conversion ratio such that the previous unit of measure is canceled out in favor of the new one. It seems like a very complex operation, but in reality it is very simple.
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For example, suppose we need to express the result of the problem under consideration in miles rather than kilometers. We know that 1 mile is roughly 1.6km, so we can convert like this:
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- 80 km × 1 mi / 1.6 km = 50 mi
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- Since the unit of measurement for kilometers appears in the denominator of the fraction representing the conversion factor, it can be simplified with that of the original result, thus obtaining the conversion in miles.
- This website provides all the tools to convert the most commonly used units of measurement.
Step 3. When necessary, replace the "Space" variable in the initial equation with the formula for calculating the total distance traveled
Objects don't always move in a straight line. In these cases it is not possible to use the value of the distance traveled by replacing it with the relative variable of the standard equation for calculating the speed. On the contrary, it is necessary to replace the variable s of the formula v = s / t with the mathematical model that replicates the distance traveled by the object under examination.
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For example, let's assume that a plane is flying using a circular path with a diameter of 20 km and traveling this distance 5 times. The aircraft in question makes this journey in half an hour. In this case we need to calculate the entire distance traveled by the aircraft before being able to determine its speed. In this example we can calculate the distance traveled by the plane using the mathematical formula that defines the circumference of a circle and we will insert it in place of the variable s of the starting equation. The formula for calculating the circumference of a circle is as follows: c = 2πr, where r represents the radius of the geometric figure. By carrying out the necessary replacements, we will obtain:
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- v = (2 × π × r) / t;
- v = (2 × π × 10) / 0.5;
- v = 62.83 / 0.5 = 125, 66 km / h.
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Step 4. Remember that the formula v = s / t is relative to the average speed of an object
Unfortunately, the simplest equation to calculate the speed we have used so far has a small "flaw": technically it defines the average speed at which an object travels. This means that the latter, according to the equation under consideration, moves at the same speed for the entire distance traveled. As we will see in the next method of the article, calculating the instantaneous speed of an object is much more complex.
To illustrate the difference between average speed and instantaneous speed, try to imagine the last time you used the car. It is physically impossible that you have been able to travel consistently at the same speed for the entire journey. On the contrary, you started from a standstill, accelerated to cruising speed, slowed down at an intersection due to a traffic light or stop, accelerated again, found yourself in a queue in traffic, etc. until you reach your destination. In this scenario, using the standard equation for the computation of velocity, all individual variations of the velocity due to normal real-world conditions would not be highlighted. Instead, a simple average is obtained of all the values assumed by the speed over the entire distance traveled
Part 3 of 3: Calculating the Instant Speed
Note:
this method uses mathematical formulas that may not be familiar to someone who has not studied advanced mathematics in school or college. If this is your case, you can broaden your knowledge by consulting this section of the wikiHow Italy website.
Step 1. Speed represents how quickly an object changes its position in space
Complex calculations related to this physical quantity can cause confusion because in mathematical and scientific fields the velocity is defined as a vector quantity composed of two parts: intensity and direction. The absolute value of the intensity represents the rapidity or speed, as we know it in everyday reality, with which an object moves regardless of its position. If we take into consideration the velocity vector, a change in its direction can also involve a change in its intensity, but not in the absolute value, that is, of the velocity as we perceive it in the real world. Let's take an example to better understand this last concept:
Let's say we have two cars that are traveling in the opposite direction, both at speeds of 50 km / h, so both are moving with the same speed. However, since their direction is opposite, using the vector definition of speed we can say that one car travels at -50 km / h while the other at 50 km / h
Step 2. In the case of a negative speed, the relative absolute value must be used
In the theoretical field, objects can have a negative speed (in case they are moving in the opposite direction from a reference point), but in reality there is nothing that can move at a negative speed. In this case the absolute value of the intensity of the vector that describes the speed of an object turns out to be the relative speed, as we perceive and use it in reality.
For this reason, both cars in the example have a real speed of 50 km / h.
Step 3. Use the derived function of position
Assuming we have the function v (t), which describes the position of an object based on time, its derivative will describe its velocity in relation to time. By simply replacing the variable t with the instant in time in which we wish to perform the calculations, we will obtain the speed of the object at the indicated moment. At this point, calculating the instantaneous speed is very simple.
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For example, assume that the position of an object, expressed in meters, is represented by the following equation 3t2 + t - 4, where t represents the time expressed in seconds. We want to find out at what speed the object under examination moves after 4 seconds, that is with t = 4. By carrying out the calculations we will obtain:
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- 3t2 + t - 4
- v '(t) = 2 × 3t + 1
- v '(t) = 6t + 1
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Substituting t = 4 we get:
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- v '(t) = 6 (4) + 1 = 24 + 1 = 25 m / s. Technically, the calculated value represents the velocity vector, but since it is a positive value and the direction is not indicated, we can say that it is the real velocity of the object.
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Step 4. Use the integral of the function that describes the acceleration
Acceleration refers to the change in the speed of an object based on time. This topic is too complex to be analyzed with due attention in this article. However, it is sufficient to know that when the function a (t) describes the acceleration of an object based on time, the integral of a (t) will describe its velocity in relation to time. It should be noted that it is necessary to know the initial velocity of the object in order to define the constant resulting from an indefinite integral.
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For example, assume that an object experiences a constant acceleration of a (t) = -30 m / s2. Let's also assume that it has an initial speed of 10 m / s. Now we need to calculate its speed at the instant t = 12 s. By performing the calculations we will get:
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- a (t) = -30
- v (t) = ∫ a (t) dt = ∫ -30dt = -30t + C
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To calculate C, we need to solve the function v (t) for t = 0. Since the initial velocity of the object is 10 m / s, we will get:
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- v (0) = 10 = -30 (0) + C
- 10 = C, so v (t) = -30t + 10
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Now we can calculate the speed for t = 12 seconds:
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- v (12) = -30 (12) + 10 = -360 + 10 = -350. Since the speed is represented by the absolute value of the intensity component of the relative vector, we can say that the examined object moves with a speed of 350 m / s.
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Advice
- Remember that practice makes perfect! Try to customize and solve the problems proposed in the article by replacing the existing values with others chosen by you.
- If you are looking for a quick and effective way to solve complex problem calculations on how to calculate the velocity of an object, you can use this online calculator to solve derivative problems or this one to solve integral calculations.
