Relative motion describes the movement of an object in a medium (space, fluid, air) that itself is moving when it is viewed by someone in a distance.

Example: a plane is moving in the air, which itself is moving and is being observed by someone in the ground. Another instance would be the observer is in an object that is moving, and they are viewing another object in motion (someone traveling in a car moving parallel to a moving train). For the second instance, simple vector additions would give the resulting relative velocity.

Below are some examples:

Because relative motion is dealing with vector quantities, they can be written in vector form:

V = A**i** + B**j** + C**k**

**V** is the vector; **A, B, and C** are the components of the vector; and **i, j, and k** denote that A is the x component, B is the y component, and C is the z component.

The equations for relative motion are the following:

Position: X_{b/a} = X_{b} – X_{a}. X_{b} and X_{a} are the absolute position of object a and b. X_{b/a} means the relative position of b with respect to a. Essentially, what is the position of object b from a’s perspective?

Velocity: V_{b/a} = V_{b} – V_{a}. V_{b} and V_{a} absolute velocities of object a and b. V_{b/a} means the relative velocity of b with respect to a. Essentially, how fast is object b moving from a’s perspective?

Acceleration: A_{b/a} = A_{b} – A_{a}. A_{b} and A_{a} the absolute velocities of object a and b. A_{b/a} means the relative acceleration of b with respect to a. In other words, what is the acceleration of b from a’s perspective?

## Recommended Steps:

1. Establish a coordinate system. If the problem is in 2 dimensions, use the standard cartessian coordinate plane (x and y direction). In the x direction, moving left is negative, and moving right is positive. In the y direction, moving upward is positive, and moving downward is negative.

2. Find the absolute position, velocity, and acceleration of both objects.

3. Add the absolute velocities of both objects with respect to their directions.

## Practice problems:

1. A motorboat traveling 5 m/s East encounters a current traveling 2.5 m/s, South. What is the resultant velocity of the motorboat?

Solution:

Object A: Motorboat is traveling 5 m/s east. Since It is traveling to the right, the velocity is positive.

V_{a}= +5_{i}

Object B: Current is traveling 2.5 m/s south. Since it is traveling downward, the velocity is negative.

V_{b}= -2.5j

The relative velocity of the motorboat, or the velocity of the boat with respect to the current, is

V_{b/a} = V_{b} – V_{a} = -2.5**j** + 5**i** or 5**i** – 2.5**j**

The resultant vector, is essentially the magnitude of the vector. The Pythagorean theorem can be used:

A visual diagram of the situation is shown below:

2. A plane can travel with a speed of 80 mi/hr east with respect to the air. Determine the resultant velocity of the plane (magnitude only) if it encounters a:

a. 15 mi/hr headwind.

b. 12 mi/hr tailwind

Solution:

Part a:

Object A: A plane is traveling 80 mi/hr east. Since it is traveling to the right, the velocity is positive. V_{a} = +80**i**

Object B: Headwind (wind traveling in the opposite direction of the plane) is traveling 15mi/hr west. Since it is traveling left, the velocity is negative. V_{b} = -15**i**

The relative velocity of the plane, or the velocity of the plane with respect to the current, is:

V_{b/a} = V_{b} – V_{a} = -15**i** + 80**i** = +65**i**

Because the motions of each object is in the x direction only, simply add the vector values. The resultant velocity of the plane (magnitude of velocity vector) is 65 mi/hr to the right.

Part b:

Object A: A plane is traveling 80 mi/hr east. Since It is traveling to the right, the velocity is positive. Va = +80i

Object B: Tailwind (wind traveling in the same direction of the plane) is traveling 12 mi/hr east. Since it is traveling right, the velocity is positive. Vb = +12i

The relative velocity of the plane, or the velocity of the plane with respect to the current, is

V_{b/a} = V_{b} – V_{a} = 12**i** + 80**i** = + 92**i**

Because the motions of each object are in the x direction only, simply add the vector values. The resultant velocity of the plane (magnitude of velocity vector) is 92 mi/hr to the right. See the diagram below:

**Reference:**

https://www.physicsclassroom.com/class/vectors/Lesson-1/Relative-Velocity-and-Riverboat-Problems