If you're seeing this message, it means we're having trouble loading external resources on our website. Show If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Today, we’re going to look at what displacement is and how it relates to “total distance”. Displacement FormulaDisplacement happens when an object moves from one location to another. The general equation for displacement is:
The most important part of displacement is noting that the path we take to get from one location to another does not matter. We only need to know where we started and where we ended up. Sometimes we call displacement an “as the crow flies” measurement, meaning a crow flies directly from one place to another using the shortest path possible. Similarly, displacement tells us the shortest distance from an initial location to a final location. Displacement Example 1Let’s look at skiing for example. A group of skiers will start at the top of the mountain. One skier really likes adventure, so she zigs and zags all around the mountain as she skis down the hill. This skier loves going as fast as possible, so he zips straight down from the ski lift to the bottom. This last skier sees that the slope is a lot steeper than she thought, so she takes the lift back down. We consider the top of the mountain the initial position and the bottom of the mountain the final position. All the skiers took different paths to get from the top of the mountain to the bottom of the mountain, but their displacement is exactly the same. Displacement Example 2Another example is Jackie Joyner-Kersee’s world record long jump. Her initial position is a line she runs to and begins her jump from. Her final position was 7.40 meters away. The jump was a straight line across the ground, but she jumped over the ground in an arc. The world record doesn’t care about how high off the ground she was; it only considers her starting and final positions. Determining Total DistanceBut what if you want total distance? A car’s odometer tells us how far the car has traveled in total. It is a measure of displacement that takes the path into account by breaking the path into tiny pieces. For example, it’s two miles to grandma’s house if we went directly through the forest on a straight path. But our car’s odometer says we traveled three miles. We have to go over a bridge and along a winding road, so the distance we’ve traveled is three miles, even though our displacement is less. Direction of DisplacementSometimes we want to specify not just the total displacement, but also the direction of the displacement. These two pieces of information, when used together, form what is called a vector. A vector gives us the magnitude and a direction. The magnitude is the total displacement and the direction is an arrow pointing from the initial location to the final location. A common example is giving a friend directions from their house to yours. Consider these directions: “Go left, then go right, then you will be at my house.” We can’t use these directions, because they don’t include a total displacement! But what if your friend said, “Go three blocks, then go four blocks, then go seven blocks.”? That’s not enough information either! There are no directions to turn! We need vectors; a direction to turn and how long to travel in that direction. ReviewBefore we go, let’s wrap up with a review question: In flyball, dogs take turns running down a track, grabbing a ball, then running back to the start, like a relay race. If the track is 100 meters long, what is the total displacement of the four-dog team?
The answer is 0 meters! Each dog starts and stops at the same location. Remember that displacement doesn’t take the path into account. Thanks for watching, and happy studying! 
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion.[1] It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point trajectory. A displacement may be identified with the translation that maps the initial position to the final position. A displacement may be also described as a relative position (resulting from the motion), that is, as the final position xf of a point relative to its initial position xi. The corresponding displacement vector can be defined as the difference between the final and initial positions: s = x f − x i = Δ x {\displaystyle s=x_{\textrm {f}}-x_{\textrm {i}}=\Delta {x}} In considering motions of objects over time, the instantaneous velocity of the object is the rate of change of the displacement as a function of time. The instantaneous speed, then, is distinct from velocity, or the time rate of change of the distance travelled along a specific path. The velocity may be equivalently defined as the time rate of change of the position vector. If one considers a moving initial position, or equivalently a moving origin (e.g. an initial position or origin which is fixed to a train wagon, which in turn moves on its rail track), the velocity of P (e.g. a point representing the position of a passenger walking on the train) may be referred to as a relative velocity, as opposed to an absolute velocity, which is computed with respect to a point which is considered to be 'fixed in space' (such as, for instance, a point fixed on the floor of the train station). For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity, which is a vector, and differs thus from the average speed, which is a scalar quantity. Rigid bodyIn dealing with the motion of a rigid body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement (displacement along a line), while the rotation of the body is called angular displacement.[citation needed] DerivativesFor a position vector
s
{\displaystyle \mathbf {s} }
These common names correspond to terminology used in basic kinematics.[2] By extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering and physics. The fourth order derivative is called jounce. See also
References
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What is referred to as the shortest distance from the initial position to the final position?
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