Distance, often given the variable “s”, is a measurement of space that is a straight line between two points. Distance can refer to the space between two immovable points (for example, a person's height is the distance from the bottom of the feet to the top of the head) or it can refer to the space between the current position of an object in motion and the initial location where the object began to move. Most distance problems can be solved by the equation s = v × t, where s is the distance, v is the average speed, and t is the time, or using s = ((x2 - x1)2 + (y2 - y1)2), where (x1, y1) and (x2, y2) are the x and y coordinates of the two points.
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Method 1 of 2: Calculating Distance with Average Speed and Time
Step 1. Find the average speed and time values
When trying to calculate the distance a moving object has traveled, there are two pieces of information that are important to this calculation: speed (or velocity) and time that the moving object has traveled. With this information, it is possible to calculate the distance traveled by the object using the formula s = v × t.
To better understand the process of using the distance formula, let's solve an example problem in this section. Let's say we are traveling down a road at 120 miles per hour (about 193 km per hour) and we want to know how far we will have covered in half an hour. Use 120 miles per hour as the value of the average velocity and 0.5 hours as the value of time, we will solve this problem in the next step.
Step 2. Multiply the average speed by the time
After knowing the average speed of a moving object and the time it has traveled, calculating the distance traveled is relatively easy. Just multiply the two values to find the answer.
- However, note that if the unit of time used in the average speed value is different from that used in the time value, you will need to change one to match. For example, if we had an average speed value measured in km per hour and a time value measured in minutes, you would need to divide the time value by 60 to convert it to hours.
- Let's finish our example problem. 120 miles/hour × 0.5 hours = 60 miles. Note that the units in the time value (hours) omit the denominator of the average speed (hours) leaving only the units of distance (miles).
Step 3. Change the equation to calculate another variable
The simplicity of the basic distance equation (s = v × t) makes it easy to use the equation to find the value of a variable other than distance. Just isolate the variable you want to find according to the basic rules of algebra, then enter the values of the other two variables to find the value of the third variable. In other words, to calculate the object's average velocity, use the equation v = s/t and to calculate the time elapsed by the object, use the equation t = s/v.
- For example, let's say we know that a car has covered 60 miles in 50 minutes, but we don't have a value for the average velocity as the object is moving. In this case, we can isolate the variable v in the basic distance equation to get v = d/t, then just divide 60 miles / 50 minutes to get the answer 1.2 miles/minute.
- Note that in the example, the answer for speed has an unusual unit (miles/minute). To get an answer in the more common miles/hour, multiply by 60 minutes/hour to get the result 72 miles/hour.
Step 4. Note that the variable “v” in the distance formula refers to the average velocity
It's important to understand that the basic distance formula offers a simplified view of the motion of an object. The distance formula assumes that an object in motion has a constant velocity - in other words, it assumes that an object in motion has a single, unchanging velocity. For abstract math problems, such as those you may encounter in an academic setting, it is sometimes still possible to model the motion of an object using this assumption. However, in real life, these examples often do not accurately reflect the movement of moving objects, which in fact can accelerate, slow down, stop, and reverse over time.
- For example, in the example problem above, we concluded that to cover 60 miles in 50 minutes, we would need to travel at 72 miles per hour. However, this is true only if traveling at one speed throughout the entire journey. For example, by traveling at 80 miles/hour for half the journey and 64 miles/hour for the remaining half, we will still cover 60 miles in 50 minutes - 72 miles/hour = 60 miles/50 minutes = ?????
- Calculus-based solutions that use derivatives are often a better choice than distance formulas for defining an object's velocity in real situations because changes in velocity are possible.
Method 2 of 2: Calculating the Distance between Two Points
Step 1. Find the two spatial coordinates of the two points
What if, instead of calculating the distance a moving object has traveled, you need to calculate the distance between two immovable objects? In such a case, the velocity-based distance formula described above will not work. Fortunately, different distance formulas can be used to easily calculate the straight line distance between two points. However, to use this formula, you will need to know the coordinates of the two points. If handling one-dimensional distances (as on a number line), the coordinates will consist of two numbers, x1 and x2. If you're handling distances in two dimensions, you'll need two values (x, y), (x1, y1) and (x2, y2). Finally, for three dimensions, you will need the value (x1, y1, z1) and (x2, y2, z2).
Step 2. Calculate the one-dimensional distance by subtracting the coordinate values of two points
Calculating a one-dimensional distance between two points when you already know the value of each point is easy. Just use the formula s = |x2 - x1|. In this formula, you subtract x1 from x2, then take the absolute value of your answer to find the distance between x1 and x2. Typically, you'll want to use the one-dimensional distance formula when the two points are on a line or number axis.
- Note that this formula uses absolute values (symbol " | |"). Absolute value only means that the value inside the symbol becomes positive if it is negative.
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For example, let's say we stop at the side of the road on a perfectly straight highway. If there is a city 5 miles in front of us and another city 1 mile behind us, how far are the two cities? If we set city 1 as x1 = 5 and city 2 as x1 = -1, we can calculate s, the distance between the two cities, in the following way:
- s = |x2 - x1|
- = |-1 - 5|
- = |-6| = 6 miles.
Step 3. Calculate the two-dimensional distance using the Pythagorean theorem
Calculating the distance between two points in two-dimensional space is more complicated than in one-dimensional, but not difficult. Just use the formula s = ((x2 - x1)2 + (y2 - y1)2). In this formula, subtract the two x-coordinates, calculate the square root, subtract the two y-coordinates, calculate the square root, then add the two results together and calculate the square root to find the distance between the two points. This formula applies to a two-dimensional plane - for example, on a regular x/y graph.
- The two-dimensional distance formula makes use of the Pythagorean theorem, which states that the length of the hypotenuse of the triangle on the right is equal to the square root of the square on the other two sides.
- For example, let's say we have two points in the x-y plane: (3, -10) and (11, 7), which represent the center of a circle and a point on the circle, respectively. To find the straight line distance between two points, we can calculate it in the following way:
- s = ((x2 - x1)2 + (y2 - y1)2)
- s = ((11 - 3)2 + (7 - -10)2)
- s = (64 + 289)
- s = (353) = 18, 79
Step 4. Calculate the three-dimensional distance by changing the two-dimensional distance formula
In three dimensions, points have z coordinates in addition to x and y coordinates. To calculate the distance between two points in three-dimensional space, use s = ((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2). This is a modified form of the two-dimensional distance formula described above that includes the z-coordinate. Subtracting the two z-coordinates, calculating the square root, and continuing with the rest of the formula ensures that your final answer will represent the three-dimensional distance between the two points.
- For example, let's say we are astronauts floating in space between two asteroids. One asteroid is about 8 km ahead, 2 km to the right, and 5 km below us, while the other is about 3 km behind, 3 km to the left, and 4 km above us. If we represent the positions of the two asteroids with the coordinates (8, 2, -5) and (-3, -3, 4), we can calculate the distance between them in the following way:
- s = ((-3 - 8)2 + (-3 - 2)2 + (4 - -5)2)
- s = ((-11)2 + (-5)2 + (9)2)
- s = (121 + 25 + 81)
- s = (227) = 15, 07 km
