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The distance, usually denoted d , is the measured length of the line connecting two points. Distance refers to the amount of space between two fixed points (for example, a person’s height is the distance from the soles of the feet to the top of the head), or the space between the current position of a moving object with its starting point. Most distance problems can be solved using the equation d = s avg × t where d is the distance, s avg is the average velocity, and t is the time, or using the equation d = √(( x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) , where (x 1 , y 1 ) and (x 2 , y 2 ) are the x and y coordinates of the two points.
Steps
Find the distance given time and average velocity
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Find the time and average velocity. When you want to find the distance an object has traveled, there are two values you need to know: its velocity and its time of motion. You can then find the distance using the formula d = s avg × t.
- To better understand the distance calculation method, consider the following example: suppose we are running on a road at 193 km/h and want to know how far we can travel in half an hour. Using 193 km/h as the average velocity value and 0.5 h as the time value, the next step is to solve the distance problem.
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Multiply the average velocity by the time. Once the average velocity and travel time of the object are known, calculating the distance traveled becomes very simple by multiplying the two values together.
- Note that if the unit of time in velocity is different from the unit of time in motion, you must convert either value to the same unit of time. For example, if we have an average velocity in km/h and a travel time in minutes, then you have to divide the time by 60 to convert it to hours.
- Let’s solve the following problem. 193 km/h × 0.5 hours = 96.5 km . Note that the unit in time value (hour) cancels out with the time unit of average velocity in the denominator (hour) so only the distance unit is km.
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Transform the equation to find other variables. Since the equation for finding the distance (d = s avg × t) is very simple, it is easy to switch sides to find variables other than distance. Keep the variable you want to find fixed and transfer the remaining variables to one side of the equation according to algebraic principles, then nest the value into two known variables to find the third variable. In other words, to find the average velocity of an object, we use the equation s avg = d/t and find the travel time using the equation t = d/s avg .
- For example, suppose a car has traveled 60 km in 50 minutes, but we do not know the average speed of the car. So we keep the variable s avg fixed in the distance equation to get the equation s avg = d/t, then divide 60 km/50 minutes to find 1.2 km/min.
- Note that the velocity found in the above problem has an uncommon unit (km/min). To get the usual speed of km/h we multiply it by 60 minutes/hour and get 72 km/h .
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The “s avg ” variable in the distance formula is the average speed. You should know that the basic distance formula above just gives us a simple view of the motion of an object. This formula assumes that the object is moving at a constant velocity , that is, it runs at a single speed over the distance to be calculated. For theoretical problems commonly encountered in school, you can sometimes simulate the motion of an object using this assumption. However, in practice, such motion simulation is not accurate because the object will increase and decrease speed, sometimes stop or reverse.
- For example, in the above problem, we assume that to cover a distance of 60 km in 50 minutes, the car must travel at 72 km/h. This is only true when the car maintains a speed of 72 km / h throughout the journey. However, if we run 80 km/h on half the journey and 64 km/h on the other half, you still get 60 km in 50 minutes, so 72 km/h is not the only result!
- Derivative solutions derived from real-world calculations are a more accurate solution to finding the velocity of real-world objects, because of the fact that the velocity is very variable.
Find the distance between two points
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Find the spatial coordinates of two points. Instead of finding the distance an object moves, how would you find the distance between two fixed points? In this case the formula for finding distance based on velocity is not helpful. Fortunately we have a separate formula for finding the length of a line joining two points. However you must know the coordinates of those two points. If you need to find the distance on a single one-dimensional line (like on a coordinate axis), the coordinates of those two points are just x 1 and x 2 . If you need to find the distance on a two-dimensional plane, you need the coordinates (x,y) for each point, i.e. (x 1 ,y 1 ) and (x 2 ,y 2 ). In three-dimensional space, the coordinates required for each point are (x 1 ,y 1 ,z 1 ) and (x 2 ,y 2 ,z 2 ).
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Find the distance on a one-way line by subtracting the coordinates of two points. Calculate the distance lying on the line connecting two points when their coordinates are known using the following simple formula d = |x 2 – x 1 | . In this formula, you subtract x 1 from x 2 , then take the absolute value to get the distance between x 1 and x 2 . Calculating the distance on a one-way line usually occurs when two points lie on a number line or a coordinate axis.
- Note that this formula uses absolute values (symbol ” | | “). Absolute value means that the number in the above notation becomes positive if it was previously negative.
- Suppose we stop the car on a perfectly straight highway. If there is a small town 5 km ahead of us and a town 1 km behind, how far apart are those two towns? If we set the coordinates for town 1 to be x 1 = 5 and town 2 to be x 1 = -1, we get the distance d between the two towns as follows:
- d = |x 2 – x 1 |
- =|-1 – 5|
- =|-6| = 6 km .
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Find the distance on a two-dimensional plane using the Pythagorean theorem. Finding the distance between two points lying in a two-dimensional plane is more complicated than a one-dimensional line, but it’s not difficult either. Use the formula d = √((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) . In this formula, you subtract the two x coordinates and then square the result, subtract the two y coordinates and square the result, then add the two results and take the square root to get distance between two points. The above formula applies to a two-dimensional plane, for example on an x/y graph.
- The formula for calculating the distance in a 2D plane uses the Pythagorean theorem, according to which the hypotenuse of a right triangle is equal to the square root of the sum of the squares of the other two sides.
- Suppose we have two points on the xy plane whose coordinates: (3, -10) and (11, 7) correspond to the center of the circle and a point lying on the circle. To find the straight distance between these two points, we solve as follows:
- d = ((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 )
- d = ((11 – 3) 2 + (7 – -10) 2 )
- d = (64 + 289)
- d = (353) = 18.79
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Find distances in 3D space by developing formulas for 2D planes. In 3D space, in addition to two coordinates x and y, the points also have z coordinates. Use the following formula to find the distance between two points in space: d = √((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 + (z 2 – z 1 ) 2 ) . This formula is derived from the formula for the plane by adding the z coordinate. Subtract the two z coordinates for each other and then square, continue to do so with the remaining two coordinates, you will surely have the distance between two points in space.
- Suppose you are an astronaut flying through space, close to two celestial bodies. One is 8 km ahead of you, 2 km to the right and 5 km below, the other is 3 km behind you, 3 km to the left and 4 km upwards. The coordinates of the two celestial bodies are as follows (8.2,-5) and (-3,-3.4), the distance between them will be:
- d = ((-3 – 8) 2 + (-3 – 2) 2 + (4 – -5) 2 )
- d = ((-11) 2 + (-5) 2 + (9) 2 )
- d = (121 + 25 + 81)
- d = (227) = 15.07 km
X
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 14 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 23,023 times.
The distance, usually denoted d , is the measured length of the line connecting two points. Distance refers to the amount of space between two fixed points (for example, a person’s height is the distance from the soles of the feet to the top of the head), or the space between the current position of a moving object with its starting point. Most distance problems can be solved using the equation d = s avg × t where d is the distance, s avg is the average velocity, and t is the time, or using the equation d = √(( x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) , where (x 1 , y 1 ) and (x 2 , y 2 ) are the x and y coordinates of the two points.
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