Distance and Displacement

Distance

The length of the actual path traversed by an object during motion in a given interval of time is called distance travelled by the object.

Suppose an object goes from A to C following path ABC, attend time t. Then total distance travelled by object in time interval t = AB + BC.

If the object goes from A to B, B to C and C to A in time t then the total distance travelled by object in time t = AB + BC + CA.

Distance is a scalar quantity. It’s value can never be zero or negative, during the motion of an object.

Displacement

The displacement of an object in a given interval of time is defined as the shortest distance between the two positions of the object in a particular direction during that time and is given by the vector drawn from the initial position to its final position.

Displacement is a vector as it possesses both magnitude and direction. When an object goal on the path ABC, then a displacement of the object is $ \displaystyle \overrightarrow{{AC}}$.

The arrowhead at C shows that the object is displaced from A to C. Increase the object is this place from C to A, then the displacement of the object is $ \displaystyle \overrightarrow{{CA}}$, (vector drawn from C to A). Then,

$ \displaystyle \overrightarrow{{AC}}=\overrightarrow{{CA}}$

It means, the displacement $ \displaystyle \overrightarrow{{CA}}$ is having the same magnitude as that of vector AC but opposite direction. In case the object goes from A to B, B to C and C to A, then the total displacement will be $ \displaystyle \overrightarrow{{AA}}=\overrightarrow{0}$.

Characteristics of displacement

1. The displacement of an object has the unit of length.
2. The displacement of an object in a given interval of time can be positive, zero or negative.
3. The magnitude of the displacement of an object between two points give the shortest distance between those two points
4. The displacement of an object between two points does not tell the type of motion followed by object between those two points. If a particle goes from A to B following shortest path or circular path, displacement in each case is $\displaystyle \overrightarrow{{AB}}$.
5. The displacement of the object between two point has a unique value.
6. The actual distance travelled by the object in the given time interval can be equal to or greater than the magnitude of the displacement.
7. The displacement of an object is unaltered due to the shift in the origin of the position axis.
8. Displacement is a single valued function of time i.e. a particle cannot be at two different positions at the same time.

Example:

A particle moves along a circle of radius R. It is starts from A and move in anticlockwise direction. Calculate the distance travelled by the particle

1. from A to B
2. from A to C
3. from A to D
4. in one complete revolution.

Also calculate the displacement in each case.

Solution:

1. Distance travelled by the particle from A to B

$ \displaystyle \begin{array}{l}=\dfrac{{2\pi r}}{4}=\dfrac{{\pi r}}{2}\\\text{Displacement }=\left| {\overrightarrow{{AB}}} \right|=\sqrt{{O{{A}^{2}}+O{{B}^{2}}}}\\=\sqrt{{{{r}^{2}}+{{r}^{2}}}}=\sqrt{2}r\end{array}$

2. For the motion form A to C, distance travelled

$ \displaystyle \text{Displacement }=\left| {\overrightarrow{{AC}}} \right|=2r$

3. For the motion A to D, distance travelled

$ \displaystyle \begin{array}{l}=\dfrac{{2\pi r\times 3}}{4}=\dfrac{3}{2}\pi r\\\text{Displacement }=\left| {\overrightarrow{{AD}}} \right|=\sqrt{{{{r}^{2}}+{{r}^{2}}}}\\=\sqrt{2}r\end{array}$

4. For one complete revolution, i.e., motion from A to A, total distance travelled = $ \displaystyle 2\pi r$.

Displacement = zero since the final position coincides with the initial position.