The distance gives the total movement of any object regardless of its direction of motion along the path; then, can distance be a curve?

**The path length between two points can be straight or curved. The direction of an object under motion does not affect the distance because the distance is a scalar quantity. It has only magnitude. Thus distance can be a curve.**

The extent of length covered by any object between two points is the distance. The distance measures the actual length between two points. Since distance is independent of the object’s direction of motion, it is not necessary to be straight. So in this post, we will learn can distance be a curve and the facts about the distance in the curve.

**Why distance is a curve?**

Distance is scalar and independent of direction of the motion which implies distance can be curve.

**The distance can be a curve based on the direction of the path traced by the object. Distance generally describes how far an object travels. When an object traces a path resembling an arc-like structure, we can say that the object traveled a curved path covering the distance.**

We already know that distance is the path traced by a body between given two points. The body has two options to cover the space between those points. Either the body has to trace a straight line from one point to another, or it must trace a curved line to reach the point. If the body chooses a curved line, it reaches the same point it is supposed to reach. So distance can be a curve, no matter what direction the body has chosen.

**How distance can be a curve?**

We know that the direction and velocity of the object under motion keep on changing along the curved path.

**Depending on the geometry of the surface and the body, if the body traces a path in which the direction of the body frequently changes while traveling between two points, the distance traced by the body is curved.**

Let us consider two points A and B. If a ball travels from A to B, the ball must trace a path to reach B from A. Let us suppose that the ball has traced a curved path and reaches point B at a certain time. Only the direction of the path traced by the ball to reach B is changed. The length does not change between points A and B. Thus, we can state distance can also be a curve.

**According to the general theory of relativity, a curved path is the shortest distance to trace for any object to travel between two points.**

**When distance is a curve?**

Irregular path for the motion of any object illustrate the change in direction of object which in general specifies the curve.

**If any object traces a circular or spherical path while traveling from one point to another, we say the distance traveled by the object is a curve. The direction of the object traveling changes its direction when the distance is a curve.**

One can observe distance as a curve if a person has to drive in turn. If the curve’s starting point is considered as the initial point of the vehicle’s motion and the end of the turning is the final point of the moving vehicle. The path length covered by the vehicle between those points is distance. Since the vehicle travels across that distance with a change in direction at every point between the starting and ending of its motion in the turning, the distance traveled by the vehicle is a curve.

**When distance is a straight line?**

The linear motion of the body with regular change in position is called as straight line.

**If any object tends to travel between two points without changing its direction along the path, the distance covered by the body is a straight line.**

When you drop a stone vertical from a certain height, the stone begins its motion from the point you dropped it and ends its motion as it reaches the ground. The stone does not change its direction throughout the action and falls straight to the ground. Now we can say that the distance is a straight line. If you draw a graph to represent the stone’s motion from a certain height to the ground, you will get a straight line representing the distance.

**How to measure distance from a curve?**

There are several ways to measure the distance from curve. Few methods are mentioned below.

**The arc length formula is one of the most accurate ways to measure the distance between two points on the curve. Arc length gives the distance between any two points on the curve. The formula is,**

Where s is the arc length, θ is the central angle, and r is the radius of the curved path.

**Distance between two points on the curve on the XY plane.**

**A graph of distance between two points can be plotted which helps to measure the length covered by an object. The curve obtained from that graph helps measure the distance. The distance between two points A and B is given by**

x_{1} and y_{1} are the coordinates at point A, and x_{2} and y_{2} are the coordinates at point B on the XY plane.

For example, let us consider two points on the XY plane which gives a curve when an object travels between those points, as shown in the figure. The value of coordinates in the figure is (3,4) and (5,7). The solution is given below.

The value of x_{1}=3, x_{2}=5, y_{1}=4, y_{2}=7.

d_{AB}=3.60 units.

**Generally, the distance between two points on the curve is measured using a string. Take length between the two points mentioned with the help of a string. Then measure the length of the string with the help of a scale that gives the distance between two points. **This is not an appropriate way of measuring the distance from the curve. And larger distance between two points cannot be measured in this method.

**Can displacement be a curve?**

Change in the position of any object is termed displacement.

**Displacement can never be a curve because displacement is a vector quantity with magnitudes and directions.**

Displacement is the shortest distance an object travels between two points. A variation in the direction of motion of the abject along the curve can be observed when an object traces curved trajectory, which is a contradiction to the definition of displacement. Thus displacement can never be a curve.

**Why can displacement not be a curve?**

We already knew that displacement is a vector whose direction must be constant throughout the action.

**On the curve, the object’s motion does not orient in a specific direction. Since displacement depends on the direction and the curved trajectory changes its direction; thus, displacement cannot be a curve.**

Consider any zig-zag motion or curved irregular path joining the two points; the direction is not fixed. The displacement is concerned with describing the separation between the two points. It joins all the points to reach the final point to give distance.

**The distance between the two points on the straight line is always equal to displacement.**

**Conclusion:**

Let us wrap up this post by stating that distance can be a curve while displacement cannot be a curve. The direction of the object during the motion does not affect the distance; thus, it can be a curve.