# Displacement and Distance: Tutorials with Examples

Examples with explanations on displacement and distance of objects moving along straight lines. More problems and their solution can be found in this site.

## Distance and Displacement Definitions

The distance is a scalar quantity (magnitude) that describes the length of the total path covered by a moving object.
The displacement is a vector quantity (magnitude and direction) that describes the difference between the final and initial positions of a moving object. It is the shortest distance moved in a certain direction. Both distance and displacement are measured in unit of lengths. (centimeters, meters, kilometers,...)

## Examples with Detailed Solutions

### Example 1

An object moves from A to D along the red path as shown below.
a) Find the total distance covered by the object
b) Find the displacement of the object Solution:
a) Using the given scale (1km per division); the total distance d is given by
d = AB + BC + CD = 2 + 5 + 2 = 9 km
b) The final and initial and positions of the moving object are used to find the displacement. The distance from A (initial position) to D (final position) is equal to AD = 5 km.
The displacement is the vector AD whose magnitude if 5 km and its direction is to the east.

### Example 2

An object moves, along a line, from point A to B to C and then back to B again as shown in the figure below.
a) Find the distance covered by the moving object.
b) Find the magnitude of the displacement of the object.
Solution:
a) The total distance d covered by the object is
d = AB + BC + CB = 5 km + 4 km + 4 km = 13 km
b) The magnitude of the displacement is equal to the distance from A (initial position) to B (final position) which is equal to 5 km.

### Example 3

An object moves from point A to B to C to D and then back to A along the rectangle shown in the figure below.
a) Find the total distance covered by the moving object.
b) Find the displacement of the object.
Solution:
a) The total distance d is equal to the perimeter of the rectangle. Using the given scale,
d = 2 AB + 2 BC = 10 + 6 = 16 km
b) Since the moving object starts at point A and finish at A, there is no change in the position of the object and therefore the displacement is equal to zero.

### Example 4

An object moves from point A to B to C along the circle as shown in the figure below.
a) Find the total distance covered by the moving object.
b) Find the displacement of the object.
Solution:
a) The total distance d is equal to half the circumference of the circle and given by
d = (1/2)(2 * Pi * 3) = 3 Pi km
b) The magnitude of the displacement D is equal to the diameter AC of the circle and is given by
D = 2 * 3 = 6 km with direction to the East

### Example 5

An object moves from point A to point B along the circle as shown in the figure below.
a) Find the total distance covered by the moving object.
b) Find the magnitude of the displacement of the object.
Solution:
a) The total distance d is equal to the quarter the circumference of the circle and given by
d = (1/4)(2 * Pi * 3) = 1.5 Pi km
b) The magnitude of the displacement D is equal to the hypotenuse AB of the right angle ABO as shown below
Use Pythagora's theorem to find AB as follows
AB2 = 32 + 32 = 18
D = AB = 3√2 km