Distance, usually represented by the variable "d", is the measure of the space in a straight line between two points. Distance can refer to the space that separates two stationary points (for example, a person's height is the distance between the sole of their foot and the top of their head) or to the space between a moving object and its starting point. movement. Most problems involving distance can be solved by the equation d = v × t, where "d" represents distance, "v" represents velocity and "t" represents time, or by the equation d = √((x2 - x1)2 + (y2 -y1)2, where (x1y1) and (x2y2) represent the x and y coordinates of the two points.
Method 1 of 2: Calculate Distance from Speed and Time
Step 1. Determine the velocity and time values
Two pieces of information are essential to calculate the distance that a given moving body has covered: its speed and the duration of that movement. From these data, it is possible to calculate the distance the object has moved through the formula d (distance) = v (speed) × t (travel time).
- To better understand the process of applying this formula, let's solve the following example. Suppose you are driving at a speed of 72 km/h and want to know how much you have walked after half an hour of travel. Considering these data, the value of v (speed) = 72 km/h and the value of t (time) = 0.5 hours.
Step 2. Multiply speed by time
Once you've determined the object's velocity value and the time it has traveled, calculating the distance it has traveled is a simple process. To do this, just multiply these two values to get the distance value.
- Pay attention to the time units of measure in the velocity value and the displacement time value. If they are different, you will need to convert one of them to continue with the resolution. For example, if speed is given in km/h and travel time is given in minutes, we could divide the time value by 60 to convert it to hours.
- Continuing the resolution of the example, we will have 72 km/h × 0.5 hours = 36 kilometers. Note that the unit of travel time (hours) is canceled with the unit in the denominator of speed (hours), leaving only the unit of distance (kilometre).
Step 3. Modify the equation to solve different types of problems
The simplicity of this equation (d = v × t) allows it to be used to calculate the values of variables other than distance. To do this, isolate the variable you want to calculate by applying the basic rules of algebra and then substitute the known values of the other two variables to arrive at the value of the third. In other words, to find the object's velocity value, use the equation v = d/t; to find the displacement time value of the object, use the equation t = d/v.
- For example, suppose a car went 6 kilometers in 12 minutes, but we don't have a speed value. In this case, we isolate the variable "v" from the distance equation and get the new equation v = d/t. Then we divide 6 km/12 minutes and arrive at 0.5 km/min.
- Note that in this example the speed value has a non-IS time unit (km/min). In order for the answer to be expressed in km/h, we must multiply it by 60 minutes/hour to then arrive at the value of 30 km/h.
Step 4. Assume that the speed "v" of the distance formula is an average speed
It's important to keep in mind that the basic distance formula offers a simplified interpretation of the object's motion. The distance formula takes into account that the object being moved has a constant speed, that is, that the body in question moves at a speed that does not change. In abstract mathematical problems (such as those found in academia), it is still possible to take this model into account. However, in real life, it does not accurately reflect how bodies move; in real situations, an object can, over time, gain or lose speed, stop or even undergo a change in its direction of displacement.
- In the previous problem, we concluded that to travel 6 km in 12 minutes, we would have to drive at a speed of 30 km/h. However, this is only true if the car's speed is kept constant throughout the entire journey. In the case of this example, if we walked halfway at a speed of 20 km/h and the other half at 60 km/h, we would still be able to walk the 6 km in 12 minutes; however, the speed would not be considered constant.
- Solutions obtained through integral calculus are generally more accurate than those obtained through the distance formula; they more accurately represent the variations in speed that occur in real-world situations.
Method 2 of 2: Calculate Distance from Two Points
Step 1. Determine the coordinates of the points x, y and/or z
What if, instead of calculating the distance an object has traveled, you need to determine the distance that separates two objects at rest? In that case, the speed-based distance formula will be useless. Fortunately, another formula can be used to easily calculate the straight-line distance between two points. However, in order to use this formula, you will need to know the coordinates of the two points in question. If the distance is in one-dimensional space (such as a number line), the coordinates of the points are simply two numbers, x1 and x2. If the distance is in two-dimensional space, two values are needed for each point, (x1y1) and (x2y2). Lastly, if the distance is in three-dimensional space, you will need three coordinates for each point, (x1y1, z1) and (x2y2, z2).
Step 2. Calculate the distance between two points in one-dimensional space
Calculating the distance between two points in a one-dimensional space is a simple task. To do this, just use the formula d = |x2 - x1|. In this formula, you must calculate the difference between x1 and x2 and then take the modulus (absolute value) of the result to find the distance between x1 and x2. You should use this formula when the colons are arranged, for example, on a line.
- Note that the formula uses the modulo symbol ("| |"). The module serves to ensure that the values within it become positive if they are negative.
- Imagine that you are standing on the side of a perfectly straight road. If there is a city 5 km to your left and another city 1 km to your right, how far is the distance between the two cities? If we call the first city x1 = 5 and the second city of x1 = -1, we can calculate the distance between them as follows:
- d = |x2 - x1|
- d = |(-1) - (5)| = |-1 - 5|
- d = |-6| = 6 kilometers.
Step 3. Calculate the distance between two points in two-dimensional space
Calculating the distance between two points in a two-dimensional space is a little more complex than in a single dimension, but it's not difficult. For this case, use d = √((x2 - x1)2 + (y2 -y1)2). In this formula, you will calculate the difference between the x coordinates of the two points, square that first result; calculate the difference between the y coordinates; square this second result; add the two results; and take the square root to finally find the distance between the two points. This formula works for two-dimensional spaces like a Cartesian plane.
- The formula for calculating a distance in two-dimensional space makes use of the Pythagorean theorem: this theorem states that the hypotenuse of a right triangle is always equal to the square root of the sum of the squares on the other two sides.
- Imagine two points on a Cartesian plane, (3, -10) and (11, 7), which represent respectively the center of a circle and a point on that circle. To find the radius of this circle, that is, the straight line that separates these two points, do the following:
- d = √((x2 - x1)2 + (y2 -y1)2)
- d = √((11 - 3)2 + [(7 - (-10)]2) = √((11 - 3)2 + (7 + 10)2)
- d = √(64 + 289)
- d = √(353) = 18, 79.
Step 4. Calculate the distance between two points in three-dimensional space
In a three-dimensional space, points have a z coordinate in addition to the x and y coordinates. In this case, to calculate the distance between two points, use the formula d = √((x2 - x1)2 + (y2 -y1)2 + (z2 - z1)2). This is a modified version of the formula shown above that includes the z coordinate. Here, you must subtract the z coordinates of the two points, square the result, and proceed with the other operations of the formula to arrive at the final result that represents the distance at the two points.
- Imagine you are an astronaut floating in space near two asteroids. The first one is about 8 kilometers ahead of you, 2 kilometers to your right, and 5 kilometers below your position; the second is 3 kilometers behind, 3 kilometers to your left and 4 kilometers above your position. If we represent the positions of the asteroids using the coordinates (8, 2, -5) and (-3, -3, 4), we can calculate the distance between them 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.