Distance: Taj Mahal is close to Delhi but far from Chennai. The tenth floor of the building is farther away from the ground floor than the first floor. What exactly do these terms “close” and “far” refer to? DISTANCE! Distance gives us an idea regarding how far or how close a place is to our location. Distance is the measure of “how much ground an object has covered during its motion.” It gives us an idea regarding the total amount the object has moved. So, this depends on the whole path travelled, not just the starting and ending points.

Define Distance

Distance is a numerical measurement of how far apart objects or points are. It is the actual length of the path travelled from one point to another. In physics, we often use distance as a reference to a physical length or an estimation based on other criteria like “ few kilometres apart”, “some miles away. However, the distance covered by an object to go from initial to final position will be the same as the distance covered by the object to go from final to initial position. Thus, distance does not depend on the direction in which a body is moving.

From the figure, The distance covered from \(A\) to \(C\), along the red path \( = AB + BC = 7\,{\rm{m}}\).
The distance covered from \(A\) to \(C\), along the black path \( = AC = 5\,\,{\rm{m}}\).

What is the Formula for Distance?

Distance travelled by an object gives the measurement of the total path covered by it throughout its motion. The distance covered by an object increases if the rate at which the object is moving increases. The distance covered by the object also increases if the object travels for a longer period. Thus, the distance travelled by an object depends on its speed and the duration(time) for which it is in motion. Therefore, the formula of distance is:

\({\rm{ Total\,distance\,covered }} = {\rm{ Speed }} \times {\rm{ time }}\)

Distance Unit

Distance is the value of the total length of a path covered by a body. The SI unit of length is the metre, and hence the SI unit of distance is also metre \((\rm{m})\).

From the formula of distance, we can see that it depends on speed and time. SI unit of speed is metre per second, and the SI unit of time is second. Thus, \({\rm{ SI\,unit\,of\,distance }} = \frac{{{\rm{ metre }}}}{{{\rm{ second }}}} \times {\rm{ second }} = {\rm{ metre }}\)

Other frequently used units of distance are:

1. Centimetre (cm): \(1\;{\rm{cm}} = {10^{ – 2}}\;{\rm{m}}\)
2. Millimetre (mm): \(1\;{\rm{mm}} = {10^{ – 3}}\;{\rm{m}}\)
3. Kilometre (km): \(1\;{\rm{km}} = {10^3}\;{\rm{m}}\)

Distance Between Two Points

The distance between any two points is the length of the line segment joining the points.

For example, if \(A\) and \(B\) are two points and \(AB = 10\;{\rm{cm}}\), the distance between \(A\) and \(B\) is \(10\;{\rm{cm }}\).

The distance between two points is the length of the line segment joining them (but this CANNOT be the length of the curve joining them).

The formula for Distance Between Two Points

Let us assume that:

\(A = \left( {{x_1},{y_1}} \right)\)

\(B = \left( {{x_2},{y_2}} \right)\)

Let the distance between \(A\) and \(B\), \(AB = d\)

Now, we will plot the given points on the coordinate plane and join them by a line.

Applying Pythagoras theorem for the \(ABC\):

\({(AB)^2} = {(AC)^2} + {(BC)^2}\)

\({d^2} = {\left( {{x_2} – {x_1}} \right)^2} + {\left( {{y_2} – {y_1}} \right)^2}\) [Values from the figure] 

Taking square root on both sides,

\(d = \sqrt {{{\left( {{x_2} – {x_1}} \right)}^2} + {{\left( {{y_2} – {y_1}} \right)}^2}} \)

This is the Distance Formula.

Distance and Displacement

Distance is how much ground is covered by an object, regardless of its starting or ending position. Thus, there is no directional component to distance measurement, making it a scalar quantity.

Displacement is the shortest distance between two points. A displacement measurement does not consider what route the object took to change position, but only where it started and where it ended. It is easiest to picture displacement by locating where the object started and drawing a straight arrow from this point to where the object stopped moving. Remember, in physics, this arrow is called a vector. Its length corresponds to the magnitude, size, movement, and arrow points in the direction of travel. This makes displacement a vector quantity because it incorporates both movements, magnitude and direction.

Distance Vs Displacement

Distance	Displacement
The complete length of the path between any two points is called distance.	Displacement is the direct length between any two points when measured along the minimum path between them.
Distance is a scalar quantity as it only depends upon the magnitude and not the direction.	Displacement is a vector quantity as it depends upon both magnitude and direction.
Distance gives detailed route information that is followed while travelling from one point to another.	As displacement refers to the shortest path, it does not give complete information on the route.
\({\rm{ Distance }} = {\rm{ Speed }} \times {\rm{ Time }}\)	\({\rm{ Displacement = Velocity }} \times {\rm{ Time }}\)
Distance can only have positive values.	Displacement can be positive, negative, and even zero.
Distance can be measured along a non-straight path.	Displacement can only be measured along a straight path.
Distance depends upon the path, i.e., it changes according to the course taken.	Displacement does not depend upon the path, and it only depends upon the initial and final position of the body.
Distance is either greater than or equal to displacement.	Displacement is either less than or equal to the distance.

Fun Facts About Distance

1. It is a scalar quantity: Distance only depends on the total length of the path, and it is independent of the direction in which the body is moving. Thus, distance only has magnitude.
2. Distance is always positive: The distance between two points can never be negative or even zero. As soon as the body starts moving, the value of distance becomes a positive non-zero quantity.
3. The distance between two points \(A\) and \(B\) can be represented as \({\rm{|A B|}}\) and if \({\rm{|B A|}}\) represents between \(B\) and \(A\), then \({\rm{|A B| = |B A|}}\)
4. Distance is equal to displacement along a straight-line path. However, along a curved way, the distance is always greater than the displacement.
5. Euclidean distance: The length of the shortest possible path through space, between two points, that could be taken if there were no obstacles (usually formalized as Euclidean distance).
6. Geodesic distance: The shortest path between two points while remaining on some surface, such as the great-circle distance along the curve of the Earth.
7. One Light-year: It is the distance light travels in one year. Light zips through interstellar space at \({\rm{186,000}}\) miles (\({\rm{300,000}}\) kilometres) per second and \({\rm{5}}{\rm{.88}}\) trillion miles (\({\rm{9}}{\rm{.46}}\) trillion kilometres) per year.

Solved Problems on Distance

Q.1. Find the distance between the two points \({\rm{(2, – 6)}}\) and \({\rm{(7,3)}}\).
Solution: Let us assume the given points to be:
\(\left( {{x_1},{y_1}} \right) = (2, – 6),\left( {{x_2},{y_2}} \right) = (7,3)\)
The formula to find the distance between two points is:
\(d = \sqrt {{{\left( {{x_2} – {x_1}} \right)}^2} + {{\left( {{y_2} – {y_1}} \right)}^2}} \)
\(d = \sqrt {{{(7 – 2)}^2} + {{(3 – ( – 6))}^2}} \)
The distance between the two points, \(d = \sqrt {106} \)

Q.2. A car is moving at a speed of \(20\;{\rm{m}}/{\rm{s}}\). Find the total distance covered by the vehicle in 15 minutes.
Solution: Speed (s) \( = 20\;{\rm{m}}/{\rm{s}}\)
Time (t) \({\rm{ = 15}}\) minutes \( = 15 \times 60 = 900\;{\rm{s}}\)
Distance covered (d) \( = {\rm{ Speed }} \times {\rm{ Time }} = 20 \times 900 = 18000\;{\rm{m}} = 18\;{\rm{km}}\)

Summary

Distance is a numerical measurement of how far apart objects or points are. It is the actual length of the path travelled from one point to another. The distance travelled by an object depends on its speed and the duration(time) for which it is in motion. Therefore, the formula of distance is: \({\rm{ Total\,distance\,covered }} = {\rm{ Speed }} \times {\rm{ time }}\).

The SI unit of length is the metre, and hence the SI unit of distance is also metre (m). It is a scalar quantity: Distance only depends on the total length of the path.

The distance between two points \(d = \sqrt {{{\left( {{x_2} – {x_1}} \right)}^2} + {{\left( {{y_2} – {y_1}} \right)}^2}} \), this is the Distance Formula.

Displacement is the shortest distance between two points. It is a vector quantity because it incorporates both movements, magnitude and direction. Displacement of an object is always less than or equal to the actual distance travelled by the object.

The distance between two points can never be negative or even zero. The value of the distance is positive and non-zero.

Frequently Asked Questions (FAQs) on Distance

Q.1. How do you use the distance formula?
Ans: The distance between two points gives the measurement of the path covered by an object between those points. If  \(\left( {{x_1},{y_1}} \right)\) be the starting point and \(\left( {{x_2},{y_2}} \right)\) be the ending point of the object, then substitute the values of the coordinates in the distance formula to calculate the distance covered by the object, i.e., \(d = \sqrt {{{\left( {{x_2} – {x_1}} \right)}^2} + {{\left( {{y_2} – {y_1}} \right)}^2}} \).

Q.2. What is distance between two points?
Ans: Distance between two points is a numerical description of how far apart two objects are in the actual sense. The distance formula, is the algebraic expression that gives the distances between pairs of points in their coordinates. For example, in three-dimensional space, the distance between the points \((a, b, c)\) and \((d, e, f)\) is equal to \(\sqrt {(a – d)^2 + (b – e)^2 + (c – f)^2} \).

Q.3. What is ‘distance’ in simple words?
Ans: Distance is how far one thing is from another thing. It is also a measure of the space between two things. It can be measured along any path. It is the total movement of an object without any regard to direction.

Q.4. What is the distance in motion?
Ans: The length of the total path as covered by an object in motion, starting from its initial position to the position it comes to rest at, is defined as distance. It is a scalar quantity and does not depend on the direction in which the object is moving in.

Q.5. Are distance and displacement the same?
Ans: No, distance is the length of the actual path covered by a particle, while the displacement is the shortest distance between two points. Therefore, the distance and displacement of a body may or may not be equal.

Q.6. Is distance scalar or vector?
Ans: The scalar quantities are the quantities that only have magnitude, while vector quantities have both direction and magnitude. Since distance is independent of the direction in which a particle moves, distance is a scalar quantity.

Q.7. What is meant by a distance?
Ans: Distance is the measurement of the path covered by an object in motion. It is the measure of “how much ground an object has covered between its start and endpoints.”.

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