Question

A man travels 1-mile due east, 5 miles due south, 2 miles due east and finally, 9 miles due north. How far is the starting point?
A. 3 miles
B. 5 miles
C. 4 miles
D. between 5 miles and 9 miles.

Answer 1

Hint:Even though the two quantities, distance and displacement have the same units and sound similar, their definitions are completely different.
If a body moves from A to B, then, distance is equal to the measurement of the entire path the body takes to move from A to B, whereas the displacement is equal to measurement of only, how far the body has moved to B, from the point A.
Also, distance is a scalar quantity with only emphasis on the magnitude and displacement is a vector quantity with emphasis over the magnitude and the direction.

Complete step-by-step answer:
Let us consider the 4 directions i.e. North, East, South and West as Cartesian coordinate directions as shown below.

Here, O is the origin, which is also, the starting point for the man.

In the first instance, he moves 1 mile towards the east. By following the direction as given above, the path traced by him, is marked in blue in the below figure.

From there, he moves 5 miles towards the south. By following the direction as given above, the path traced by him, is marked in blue in the below figure.

.


From there, he moves 2 miles towards the east. By following the direction as given above, the path traced by him, is marked in blue in the below figure.

From there, he moves 9 miles towards the north. By following the direction as given above, the path traced by him, is marked in blue in the below figure.

The red arrow in the figure represents his shortest distance from the origin which is called the displacement which represents how far he is from the origin.
By taking the Cartesian coordinates of the movements of the man along the respective directions,
The net distance he moved along X-axis = $ + 1 + 2 = 3$ miles
The net distance he moved along Y-axis = $ - 5 + 9 = 4$ miles
Therefore, the man has totally moved 3 miles in the X-direction and 4 miles in the Y-direction as shown in the figure.
The total displacement can be found by using the Pythagoras theorem:
Net displacement, ${d^2} = {x^2} + {y^2}$ where x and y are the total displacements along the X-Y axes.
Therefore, displacement
$d = \sqrt {{x^2} + {y^2}} = \sqrt {{3^2} + {4^2}} = \sqrt {25} = 5$ miles
The man is 5 miles far from his starting point.

Hence, the correct option is Option B.

Note: The students have to make sure that they are considering the right signs and axes for the main directions because jumbling them will result in a different and incorrect answer. Hence, the students are advised to fix a standard as discussed: i) If the movement is North, take it as +Y ii) If the movement is East, take it as +X iii) If the movement is South, take it as -Y iv) If the movement is West, take it as -X