How to Find Resultant Displacement in Physics

How to Find Resultant Displacement in Physics
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The concept of displacement can be tricky for many students to understand when they first encounter it in a physics course. In physics, displacement is different from the concept of distance, which most students have previous experience with. Displacement is a vector quantity, so it has both magnitude and direction. It is defined as the vector (or straight line) distance between an initial and final position. The resultant displacement therefore depends only on knowledge of these two positions.

TL;DR (Too Long; Didn't Read)

To find the resultant displacement in a physics problem, apply the Pythagorean formula to the distance equation and use trigonometry to find the direction of movement.

Determine Two Points

Determine the position of two points in a given coordinate system. For example, assume an object is moving in a Cartesian coordinate system, and the initial and final positions of the object are given by the coordinates (2,5) and (7,20).

Set up Pythagorean Equation

Use the Pythagorean theorem to set up the problem of finding the distance between the two points. You write the Pythagorean theorem as


where c is the distance you're solving for, and x2-x1 and y2-y1 are the differences of the x, y coordinates between the two points, respectively. In this example, you calculate the value of x by subtracting 2 from 7, which gives 5; for y, subtract the 5 in the first point from the 20 in the second point, which gives 15.

Solve for Distance

Substitute numbers into the Pythagorean equation and solve. In the example above, substituting numbers into the equation gives


Solving the above problem gives c = 15.8. This is the distance between the two objects.

Calculate the Direction 

To find the direction of the displacement vector, calculate the inverse tangent of the ratio of the displacement components in the y- and x-directions. In this example, the ratio of the displacement components is 15÷5 and calculating the inverse tangent of this number gives 71.6 degrees. Therefore, the resultant displacement is 15.8 units, with a direction of 71.6 degrees from the original position.

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