Calculating Displacement: Formula and Methodology

Displacement refers to the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction. The formula for calculating displacement depends on whether the motion is one-dimensional or two-dimensional.

One-Dimensional Displacement:

For motion along a straight line (one-dimensional motion), the formula for displacement ($\Delta x$) is given by:

$\Delta x = x_f - x_i$

Where:

• $\Delta x$ is the displacement.
• $x_f$ is the final position.
• $x_i$ is the initial position.

Two-Dimensional Displacement:

For motion in two dimensions (e.g., horizontal and vertical directions), the displacement ($\mathbf{\Delta r}$) is a vector quantity and can be calculated using the Pythagorean theorem and trigonometric functions. The formula is:

$\mathbf{\Delta r} = \sqrt{(\Delta x)^2 + (\Delta y)^2}$

Where:

• $\mathbf{\Delta r}$ is the displacement vector.
• $\Delta x$ is the horizontal displacement.
• $\Delta y$ is the vertical displacement.

Additionally, the direction of the displacement vector ($\theta$) can be found using trigonometric functions:

$\theta = \tan^{-1}\left(\frac{\Delta y}{\Delta x}\right)$

Example:

Let's say an object starts at position $(2, 3)$ and ends at position $(6, 8)$. To calculate the displacement in two dimensions:

$\Delta x = 6 - 2 = 4$

$\Delta y = 8 - 3 = 5$

$\mathbf{\Delta r} = \sqrt{(4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41}$

$\theta = \tan^{-1}\left(\frac{5}{4}\right)$

So, the displacement is $\sqrt{41}$ units at an angle of $\tan^{-1}\left(\frac{5}{4}\right)$ with respect to the horizontal.

Keep in mind that displacement is different from distance traveled. Displacement considers the change in position from the initial to the final point, while distance traveled is the total length of the path taken.

d = Δx = x_f - x_i

d = Δx = x_f - x_i

d = Δx = x_f - x_i = (5, 7) - (2, 3) = (3, 4)

||d|| = √(3^2 + 4^2) = √25 = 5

θ = arctan(4/3) ≈ 53°