In physics, displacement indicates the change in position of an object. When you calculate it, you measure how far a body is "out of place" from its starting position. The formula used to calculate the displacement depends on the data provided by the problem. Methods for doing this are described in this tutorial.
Steps
Part 1 of 5: Resulting Displacement
Step 1. Apply the resulting displacement formula when using distance units to specify the start and end position
Although distance is a different concept than displacement, the resulting displacement problems specify how many "meters" an object has moved from its starting position.
- The formula in this case is: S = √x² + y². Where "S" is the displacement, x the first direction towards which the object moves and y the second. If the body moves only in a single direction, then y is equal to zero.
- An object can move in a maximum of two directions, since the movement along the north-south or east-west axis is considered a neutral movement.
Step 2. Connect the points that determine the various positions of the body and indicate them in sequential order with the letters of the alphabet from A to Z
Use a ruler to draw straight lines.
- Also remember to connect the first point with the last with a single segment. This is the displacement you need to calculate.
- For example, if an object has moved 300 meters east and 400 meters north, the segments will form a triangle. AB forms the first leg of the triangle and BC will be the second. AC, the hypotenuse of the triangle, is equal to the resulting displacement of the object. The directions of this example are "east" and "north".
Step 3. Enter the directional values of x² and y²
Now that you know the two directions in which the body moves, enter the values in place of the respective variables.
For example, x = 300 and y = 400. The formula will be: S = √300² + 400²
Step 4. Perform the calculations of the formula respecting the order of operations
First do the powers by squaring 300 and 400, then add them together and finally do the square root of the sum.
For example: S = √90.000 + 160.000. S = √250.000. S = 500. Now you know that the displacement is 500 meters
Part 2 of 5: Known Speed and Time
Step 1. Use this formula when the problem tells you the speed of a body and the time it takes
Some physics problems do not give the distance value, but they say how long an object has moved and at what speed. Thanks to these values you can calculate the displacement.
- In this case the formula is: S = 1/2 (u + v) t. Where u is the initial velocity of the object (or the velocity possessed when the movement is considered); v is the final speed, that is the one possessed once the destination has been reached; t is the time taken to travel the distance.
- Here is an example: a car travels on the road for 45 seconds (time considered). He turned west at a speed of 20 m / s (initial speed) and at the end of the route his speed was 23 m / s. Calculate the displacement based on these factors.
Step 2. Enter the speed and time data by replacing them with the appropriate variables
Now you know how long the car has traveled, its initial speed, its final speed and therefore you can trace its displacement from the starting point.
The formula will be: S = 1/2 (20 m / s + 23 m / s) 45 s
Step 3. Perform the calculations
Remember to follow the order of operations, otherwise you will get a completely wrong result.
- For this formula, it does not matter whether you reverse the initial speed with the final one. Since the values will be added, the order does not interfere in the calculations. For other formulas, on the other hand, inverting the initial speed with the final one involves different displacements.
- Now the formula should be: S = 1/2 (43 m / s) 45 s. First you divide 43 by 2, getting 21.5. Finally multiply the quotient by 45 and you get 967.5 meters. This corresponds to the displacement value, i.e. how much the car has moved with respect to the starting point.
Part 3 of 5: Known Velocity, Acceleration and Time
Step 1. Apply a modified formula when, in addition to the initial speed, you also know the acceleration and time
Some problems will only tell you the initial speed of a body, the travel time and its acceleration. You will need to use the equation described below.
- The formula you need to use is: S = ut + 1 / 2at². "U" represents the initial speed; "a" the acceleration of the body, that is, how quickly its speed changes; "t" is the total time considered or even a certain period of time in which the body has accelerated. In both cases it will identify itself with the normal units of time (seconds, hours and so on).
- Suppose a car travels at 25m / s (initial speed) and starts accelerating at 3m / s2 (acceleration) for 4 seconds (time). What is the movement of the car after 4 seconds?
Step 2. Enter your data into the formula
Unlike the previous one, only the initial speed is represented, so be careful not to make a mistake.
Considering the previous example, the equation should look like this: S = 25 m / s (4s) + 1/2 (3 m / s²) (4s) ². The use of parentheses helps you to keep the time and acceleration values separate
Step 3. Calculate the displacement by performing the operations in the right order
There are many mnemonic tricks to remember this order, the most famous being the one in English PEMDAS or " P.lease Andxcuse my dear TOunt S.ally "where P stands for parentheses, E for exponent, M for multiplication, D for division, A for addition and S for subtraction.
Read the formula: S = 25 m / s (4s) + 1/2 (3 m / s²) (4s) ². First, square 4 and you get 16. Then multiply 16 by 3 to get 48. Proceed to multiply 25 by 4 which gives you 100. Finally divide 48 by 2 to get 24. Your simplified equation looks like: S = 100 m + 24 m. At this point you just have to add the values, and you find the total displacement equal to 124 m
Part 4 of 5: Angular displacement
Step 1. When an object follows a curved path, you can calculate the angular displacement
Although in this case you consider moving along a straight line, you need to know the difference between the final and starting position when the moving body defines an arc.
- Think of a little girl sitting on the merry-go-round. As it spins around the outer edge of the carousel, it defines a curved line. Angular displacement measures the minimum distance between the start and end position of an object that does not follow a straight path.
- The formula for angular displacement is: θ = S / r, where "S" is the linear displacement, "r" is the radius of the defined portion of the circumference and "θ" is the angular displacement. The value of S is the displacement along the circumference of a body, the radius is the distance between the body and the center of the circumference. Angular displacement is the value we are looking for.
Step 2. Enter the radius and linear displacement data into the formula
Remember that the radius is the distance from the center of the circumference to the moving body; sometimes you may be given the diameter, in which case just divide it by two to get the radius.
- Here's a simple problem: a little girl is on the moving merry-go-round. She is sitting 1 meter from the center of the carousel (radius). If the girl moves along an arc of 1.5m (linear displacement), what will the angular displacement be?
- Your equation, once you have entered the data, will be: θ = 1, 5 m / 1 m.
Step 3. Divide the linear displacement by the radius
By doing this you find the angular displacement.
- By performing the calculation you get that the girl has undergone a shift of 1, 5 radians.
- Since angular displacement calculates how far a body has turned from its initial position, it must be expressed as an angle and not as a distance. Radians are the unit of measurement for angles.
Part 5 of 5: Concept of Displacement
Step 1. Remember that "distance" has a different meaning than "displacement"
The distance refers to the length of the entire path traveled by an object.
- Distance is a "scalar magnitude" and takes into account the entire path taken by an object without considering the direction in which it traveled.
- For example, if you walk 2 meters to the east, 2 meters to the south, 2 to the west and finally 2 to the north, you will find yourself in the original position. Although you have traveled one distance of 8 meters, yours shift is zero, since you find yourself at the starting point (you followed a square path).
Step 2. Remember that displacement is the difference between two positions
It is not the sum of the distances traveled, but only focuses on the starting and ending coordinates of a moving body.
- The displacement is a "vector quantity" and expresses the change in position of an object considering also the direction in which it moved.
- Let's say you move east for 5 meters. If you then go back west for another 5 meters, you travel in the opposite direction from the beginning. Even though you walked 10 meters, you have not changed your position and your displacement is 0 meters.
Step 3. Remember the words "back and forth" when imagining the shift
Moving in the opposite direction cancels the movement of an object.
Imagine a football manager walking back and forth along the sideline. As he shouts instructions to the players, he moves from left to right (and vice versa) many times. Now imagine he stops at a point on the sideline to talk to his team captain. If it is in a different position than the initial one, then you can see the movement made by the coach
Step 4. Remember that displacement is measured along a straight, not curved line
To find the displacement you need to find the shortest and most efficient path that connects the starting position to the final one.
- A curved path will take you from your original location to your destination, but this is not the shortest route. To help you visualize this, imagine walking in a straight line and encountering a pillar. You can't cross this obstacle, so you bypass it. Eventually you will find yourself in a spot identical to the one you would have occupied if you could have "crossed" the pillar, but you had to take extra steps to get there.
- Although the displacement is a rectilinear quantity, know that you can also measure the displacement of a body that follows a curved path. In this case we speak of "angular displacement" and is calculated by finding the shortest trajectory that leads from the origin to the destination.
Step 5. Remember that displacement can also be a negative number, unlike distance
If to get to your final destination you had to move in a direction opposite to that of departure, then you have moved a negative value.
- Let's consider the example where you walk 5 meters to the east and then three to the west. Technically you are 2m from your original position and your displacement is -2m because you have moved in opposite directions. However, the distance is always a positive value because you cannot "non-move" for a certain number of meters, kilometers and so on.
- A negative shift does not indicate that it has decreased. It simply means that it happened in the opposite direction.
Step 6. Keep in mind that sometimes distance and displacement can be the same thing
If you walk in a straight line for 25 meters and then stop, the length of the journey you have traveled is equal to the distance you are from the starting point.
- This only applies when you move from the origin in a straight line. Let's say you live in Rome, but you have found a job in Milan. You have to move to Milan to be close to your office and then take a plane that takes you directly there covering 477 km. You traveled 477km and moved 477km.
- However, if you had taken the car to move, you would have traveled 477 km but you would have covered a distance of 576 km. Since driving on the road forces you to change direction to get around orographic obstacles, you will have traveled a longer route than the shortest distance between the two cities.
