Distance, often denoted by the variable d, is a measure of space indicated by a straight line connecting two points. Distance can refer to the space between two stationary points (for example, a person's height is the distance from the tip of his toes to the top of his head) or it can refer to the space between a moving object and the its initial position. Most distance problems can be solved with the equation d = s × t where d is the distance, s the speed and t the time, or da d = √ ((x2 - x1)2 + (y2 - y1)2, where (x1, y1) and (x2, y2) are the x, y coordinates of two points.
Steps
Method 1 of 2: Finding the Distance with Space and Time
Step 1. Find the values for space and time
When we are trying to calculate the distance that a moving object has traveled, two pieces of information are fundamental to carry out the calculation, it is possible to calculate this distance with the formula d = s × 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 a road at 120 miles per hour (about 193 km / h) and we want to know how far we have traveled if we have traveled for half an hour. Using 120 mph as a value for the speed e 0.5 hours as a value for time, we will solve this problem in the next step.
Step 2. We multiply the speed and time
Once you know the speed for a moving object and the time it has traveled, finding the distance it has traveled is fairly simple. Just multiply these two quantities to find the answer.
- Note, however, that if the units of time used in the value of your speed are different from those used in the value of time, you will have to convert one or the other in order to make them compatible. For example, if we had a speed measured in km / h and a time measured in minutes, we would have to divide the time by 60 to convert it into hours.
- Let's solve our example problem. 120 miles / hour × 0.5 hours = 60 miles. Note that the units in the value of time (hours) are simplified with the unit in the denominator of the speed (hours) to leave only one unit of distance measurement (miles)
Step 3. Flip the equation to find the values of the other variables
The simplicity of the basic distance equation (d = s × t) makes it quite easy to use the equation to find the values of other variables beyond the distance. Simply isolate the variable you want to find based on the rules of algebra, then enter the value of the other two variables to find the value of the third. In other words, to find the speed, use the equation s = d / t and to find the time you traveled for, use the equation t = d / s.
- For example, let's say we know that a car has traveled 60 miles in 50 minutes, but we don't know the value of its speed. In this case, we can isolate the variable s in the basic distance equation to get s = d / t, then we simply divide 60 miles / 50 minutes to get the answer equal to 1.2 miles / minute.
- Note that in our example, our response for speed has an uncommon unit of measurement (miles / minutes). To express our answer in the form of miles / hour, we want to multiply it by 60 minutes / hour to get 72 miles / hour.
Step 4. Note that the "s" variable in the distance formula refers to the average speed
It is important to understand that the basic distance formula offers a simplistic view of the movement of an object. The distance formula assumes that the moving object has a constant speed; in other words, it assumes that the object is moving at a single speed, which does not vary. For an abstract mathematical problem, such as those in the academic field, in some cases it is possible to model the motion of an object starting from this assumption. In real life, however, it often does not accurately reflect the movement of objects, which can increase, decrease their speed, stop and go back in some cases.
- For example, in the previous problem, we concluded that to travel 6 miles in 50 minutes, we would have to travel at 72 miles / hour. However, this is only true if we could travel at that speed all the way. For example, traveling at 80 miles / hour for half the route and 64 miles / hour for the other half, we would always have traveled 60 miles in 50 minutes.
- Solutions based on analysis such as derivatives are often a better choice than the distance formula to define the speed of an object in real world situations where the speed is variable.
Method 2 of 2: Find the Distance Between Two Points
Step 1. Find two points with x, y and / or z coordinates
What should we do if, instead of finding the distance traveled by a moving object, we had to find the distance of two stationary objects? In cases like these, the speed-based distance formula would be of no help. Fortunately, another formula can be used that allows you to easily calculate the distance in a straight line between two points. However, to use this formula, you will need to know the coordinates of the two points. If you are dealing with a one-dimensional distance (such as on a numbered line), the coordinates of your points will be given by two numbers, x1 and x2. If you are dealing with a two-dimensional distance, you will need the values for two points (x, y), (x1, y1) and (x2, y2). Finally, for three-dimensional distances, you will need values for (x1, y1, z1) and (x2, y2, z2).
Step 2. Find the 1-D distance by subtracting the two points
Calculating the one-dimensional distance between two points when you know the value of each is a breeze. It is enough to use the formula d = | x2 - x1|. In this formula, subtract x1 from x2, then take the absolute value of the result to find the solution x1 and x2. Typically, you will use the one-dimensional distance formula if your points are on a straight line.
- Note that this formula uses the absolute value (the symbol " | |"). The absolute value implies that the term contained within it becomes positive if it were negative.
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For example, suppose we stopped at the side of a perfectly straight road. If there is a small town 5 miles ahead and one 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 find d, the distance between the two cities, as:
- d = | x2 - x1|
- = |-1 - 5|
- = |-6| = 6 miles.
Calculate Distance Step 7 Step 3. Find the 2-D distance using the Pythagorean Theorem
Finding the distance between two points in two-dimensional space is more complicated than it was in the one-dimensional case, but it's not difficult. Just use the formula d = √ ((x2 - x1)2 + (y2 - y1)2). In this formula, you subtract the x coordinates of the two points, square, subtract the y coordinates, square, add the two results together, and take the square root to find the distance between your two points. This formula works as in the two-dimensional plan; for example, on x / y charts.
- The 2-D distance formula uses the Pythagorean Theorem, which says that the hypotenuse of a right triangle is equal to the sum of the squares of the legs.
- For example, suppose we have two points on the x / y plane: (3, -10) and (11, 7) representing the center of a circle and a point on the circle, respectively. To find the straight line distance between these two points, we can proceed as follows:
- d = √ ((x2 - x1)2 + (y2 - y1)2)
- d = √ ((11 - 3)2 + (7 - -10)2)
- d = √ (64 + 289)
- d = √ (353) = 18.79
Calculate Distance Step 8 Step 4. Find the 3-D distance by modifying the 2-D case formula
In three dimensions, the points have an additional z coordinate. To find the distance between two points in three-dimensional space, use d = √ ((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2). This is the 2-D distance formula modified to take the z coordinate into account as well. Subtracting the z coordinates from each other, squaring them, and proceeding as before over the rest of the formula, will ensure that the final result represents the three-dimensional distance between two points.
- For example, suppose you are an astronaut who is floating in space near two asteroids. One is about 8km in front of us, 2km to the right and 5km below, while the other is 3km behind us, 3km to the left and 4km above us. If we represent the position of these two asteroids with the coordinates (8, 2, -5) and (-3, -3, 4), we can find the mutual distance of the two asteroids as follows:
- 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
