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### How to find distance and displacement with detailed solutions

Problems and Solutions about distance and displacement are presented and updated.

**Problem 1: **

An object moves from point A to B, C, and D finally, along a rectangle.

(a) Find the magnitude and direction of the displacement vector of the object?

(b) Find the distance traveled by that object?

(c) Suppose the object returns to the point A, its initial position. Now, Find the displacement and distance?

**Solution:**

(a) By definition of displacement, connect the initial (A) and final (D) points together. As shown, displacement is toward the negative of the y-axis and its magnitude is equal to the width of the rectangle.

(b) distance is equal to the length of AB+BC+CD.

(c) Since the initial and final positions are the same, so by definition of displacement, the difference between them is zero. But distance, in this case, is the perimeter of the rectangle.

**Problm 2:**

(a) Find the magnitude and direction of the displacement vector?

(b) How much distance traveled by this moving object?

(c) Suppose the object returns to the point A, its initial position. Now, Find the displacement and distance?

**Solution 2:**

(a) Connect the initial (A) and final (C) points together by the straightest line. The magnitude and direction of that line represent the displacement vector. Here, the displacement is the hypotenuse of the right triangle. So using the Pythagorean theorem, one finds its magnitude as

\begin{align*}

c^2 &= a^{2} + b^{2} \\

\Rightarrow c&=\sqrt{a^{2} +b^{2}}\\

&= \sqrt{3^{2} + 4^{2}}=5\,{\rm m}

\end{align*}

the direction of displacement is shown in figure (to the northeast).

(b) Adding the base and height of this triangle gets the distance traveled from (A) to (C).

\begin{align*}

\text{distance} &= a + b\\

&= 3 + 4\\

&= 7\,{\rm m}

\end{align*}

(c) Displacement is zero since the object returns to the initial position. Distance is also computed as the perimeter of the triangle.

**Important notes about displacement and distance:**

- If the initial and final points are the same, so the displacement is zero.
- Distance is measured as the perimeter of the path traveled by the moving object.

For more involved and difficult problems with a thorough explanation about the definition of displacement in two and three dimensions visit the exam center or course pages.

In addition, you can also check out the Wikipedia article about distance and displacement.