Position, Distance, and Displacement

The location of an object at rest pertains to its position. As the object moves from one place to another, it covers a distance. The total distance is the sum of all the distances travelled by an object from one location to another. Distance is considered as a scalar quantity, described by a magnitude and a unit only. When the distance from the initial position straight to the final position of the object, the displacement is measured. It is defined by a magnitude, unit and the direction of the motion thus, displacement is a vector quantity. The figure shows an illustration of the three quantities.

The object as represented by the dot is at rest at its position. The total measure from its position going to East then to North is the distance covered by the object. The straight way from the initial to final position gives the displacement of the object. The angle $$\theta$$ is the direction of the displacement.

Example 1.

Mitch walks from home to school. She has walked $$5\;m$$ already when she found out that she left her physics project. So, she went back and after $$2 \;m$$ of walking, she met her mother bringing her project. And so, she went her way again to school. If the school is $$10\;m$$ away from her home, how long does she walked?

Solution:

To solve how long does Mitch walk during the entire situation, we will just simple add the distances to get the total distance.

$$d=5\;m+2\;m+7\;m=14\;m$$

Example 2.

Izrafel and Imanuel are planning to have a swim on the river nearby. They are planning to have a race, Imanuel to take the road and Izrafel to take a short-cut way to the river. If the way to the river from their house is $$100\;m\;\text{North}$$ and $$200\;m\;\text{East}$$, who will get to the river first?

Solution 1:

For Imanuel, we will be calculating the distance he traveled from home to the river.

$$d= 100\;m + 200\;m = 300\;m$$

Solution 2.

SInce Izrafel is taking a way straight from home to the river, this refers to the displacement. We will plot the given vectors to form a right triangle, and solve for the displacement using the Pythagorean Theorem.

$$d=\sqrt{(100\;m)^2+(200\;m)^2}=\sqrt{50,000\;m^2}=223.61\;m$$

To solve for the direction of the displacement, we have

$$\tan \theta=\frac {200\;m}{100\;m}=2$$

$$\theta=tan^{-1}\;2=63.43^\circ\;\text{East of North}$$

Thus, Izrafel traveled $$223.61\;m$$$$63.43^\circ\;\text{East of North}$$.

Solution 3.

Based on the results, Izrafel will arrive first.