 # Distance and Displacement

Distance and displacement are two quantities that appear to be the same to many people but are in reality quite different from each other. Furthermore, distance refers to the amount of ground an object covers during its motion. In contrast, displacement is the measure that tells us how far an object is out of place.

## Introduction to Distance and Displacement

Distance refers to an object’s total movement irrespective of its direction. In contrast, displacement is the change which takes place in the position of an object. As such, simply speaking, the most important distance and displacement difference is that distance is a scalar quantity while displacement is a vector quantity.

One can define distance as the total length of the path on which travelling takes place from one location to another location.  In contrast, displacement is the minimum distance possible between a body’s initial and final positions. Distance and Displacement

### How do we Measure Distance and Displacement?

For Distance:

One must follow the below procedure in order to measure the distance

• Place the pegs to mark the start and end of the distance whose measurement is to take place.
• Hold the zero point of the tape or chain at the starting peg’s centre.
• Drag the tape or chain in the second peg’s direction. Furthermore, make sure to pull the tape or chain straight before measuring.
• Any knots or entangled links can cause errors in the measurement.
• When using a measuring tape, one can directly read the distance between the two pegs.
• When using a chain, one must count the number of links between the two pegs. Most noteworthy, the total distance would be equal to the number of links multiplied by one link’s length.

For displacement:

In physics, one can calculate displacement by calculating the distance that is present between the initial position and the final position of an object.

Consider the example of a fine new golf ball that can roll around. This golf ball tends to roll around on top of a large measuring stick. In order to calculate displacement, place the ball at the 0 position on the measuring stick.

When the golf ball rolls over to a new point, displacement takes place. Therefore, if the ball moves 3 meters to the right from its initial position of 0 meters, the displacement taking place shall be 3 meters.

### Formula of Distance and Displacement

For distance:

The formula is Δd = d1 + d2

For displacement:

Displacement refers to the change in an object’s position from the origin. Since it is a vector quantity, it involves both magnitude and direction.

Displacement formula will be = (final position) – (initial position) = change in position

D = Xf -Xi

D = displacement

Xf = final position

Xi = initial position

ΔX = short form for the change which takes place in position

### Derivation of the Formula of Distance and Displacement

Derivation of distance formula:

Let P(x1, y1) and Q(x2, y2) be the coordinates of two points that are present on a coordinate plane.

Draw two lines that are parallel to both x-axis and y-axis through the points P and Q.

The parallel line through P must come into contact with the perpendicular drawn to the x-axis from Q at T.

Thus, ΔPTQ is right-angled at point T.

PT = Base, QT = Perpendicular and PQ = Hypotenuse

Furthermore, applying Pythagoras Theorem,

PQ2 = PT2 + QT2

= (x2 – x1)2 + (y2 – y1)2

Moreover, PQ = √[(x2 – x1)2 + (y2 – y1)2]

Hence, the distance that is present between two points (x1, y1) and (x2, y2) is √[(x2 – x1)2 + (y2 – y1)2]

Similarly, in the Cartesian plane, the expression of the distance of a point P(x, y) from the origin O(0, 0) can take place by the following formula:

OP = √(x2 + y2)

Derivation of displacement formula:

Since, Displacement(s)= velocity(v)*time(t) .

Therefore, its derivation can take place as

s = ut+1/2at2, where u represents the initial velocity,

Moreover, a is the acceleration while t represents the time.

Making use of a graphical method, consider a velocity vs time graph. Here

Point A(u,t1), B (v,t2).

Area of the graph (i.e. v*t ) would provide the displacement taking place during the time interval.

Therefore, the area of ABCDE  = area of triangle ABC + Area of rectangle ACDE.

=1/2*AC*BC + (ED*AE)

=1/2*t ‘*(v-u) + u*t ‘ {t2-t1=t and v is the final velocity derived}

Now one can use 1st equation of motion, v – u = a*t ‘ ;

1/2* a*(t’)2 +u* t ‘ // is the relevant equation.

## FAQs For Distance and Displacement

Question 1: What is displacement in physics or science?

Answer 1: In physics, displacement refers to the change that happens in the position of an object. Furthermore, displacement happens to be a vector quantity.

Question 2: Differentiate between scalar and vector quantity?

Answer 2: A scalar quantity comprises of only a magnitude. In contrast, a vector quantity is characterized by a magnitude as well as a direction.

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