Relative Velocity - General

Relative velocity is a fundamental concept in kinematics, which is a branch of classical mechanics that describes the motion of points, bodies, and systems of bodies without considering the forces that cause them to move. Relative velocity refers to the velocity of one object as observed from another moving object. Understanding relative velocity is crucial for solving problems involving multiple moving objects, such as vehicles on a road, boats in a river, or aircraft in the sky.

Understanding Relative Velocity

To understand relative velocity, consider two objects, A and B, moving with velocities $\vec{v}A$ and $\vec{v}_B$ respectively. The relative velocity of A with respect to B is the velocity that A appears to have when observed from B. It is denoted as $\vec{v}{A/B}$ and is given by the formula:

$$ \vec{v}_{A/B} = \vec{v}_A - \vec{v}_B $$

Similarly, the relative velocity of B with respect to A is:

$$ \vec{v}_{B/A} = \vec{v}_B - \vec{v}_A $$

Note that $\vec{v}{A/B}$ and $\vec{v}{B/A}$ are equal in magnitude but opposite in direction.

Key Points and Differences

Aspect	Description
Frame of Reference	Relative velocity depends on the observer's frame of reference.
Vector Quantity	Relative velocity is a vector, which means it has both magnitude and direction.
Independence of Motion	The motion of one object is independent of the motion of the other.
Relative Speed	The magnitude of relative velocity is known as relative speed.
Direction of Motion	The direction of relative velocity is the direction of one object as seen from the other.

Formulas

In one dimension, where objects move along a straight line, the relative velocity can be simplified to:

$$ v_{A/B} = v_A - v_B $$

In two or three dimensions, relative velocity involves vector subtraction:

$$ \vec{v}_{A/B} = \vec{v}_A - \vec{v}_B $$

If the objects are moving at an angle to each other, the relative velocity can be found using vector components or graphical methods.

Examples

Example 1: Cars on a Highway

Two cars, A and B, are moving on a highway. Car A is moving at 60 km/h, and Car B is moving at 80 km/h in the same direction. The relative velocity of Car A with respect to Car B is:

$$ v_{A/B} = v_A - v_B = 60\ \text{km/h} - 80\ \text{km/h} = -20\ \text{km/h} $$

This means that to an observer in Car B, Car A appears to be moving backwards at 20 km/h.

Example 2: Boats in a River

A boat A is moving across a river at a velocity of 5 m/s due east, and the river's current (boat B) is flowing due north at 3 m/s. The relative velocity of the boat with respect to the river is:

$$ \vec{v}_{A/B} = \vec{v}_A - \vec{v}_B = 5\ \hat{i} - 3\ \hat{j}\ \text{m/s} $$

Here, $\hat{i}$ and $\hat{j}$ are unit vectors in the east and north directions, respectively. The relative velocity is a vector pointing northeast, and its magnitude can be found using the Pythagorean theorem:

$$ |\vec{v}_{A/B}| = \sqrt{5^2 + 3^2} = \sqrt{34}\ \text{m/s} $$

Example 3: Aircraft in the Sky

An aircraft A is flying north at a speed of 200 km/h, and a wind (aircraft B) is blowing from the west at 50 km/h. The relative velocity of the aircraft with respect to the wind is:

$$ \vec{v}_{A/B} = \vec{v}_A - \vec{v}_B = 200\ \hat{j} - 50\ \hat{i}\ \text{km/h} $$

The aircraft's actual velocity is northeastward, and its magnitude is:

$$ |\vec{v}_{A/B}| = \sqrt{200^2 + 50^2} = \sqrt{42500}\ \text{km/h} \approx 206.2\ \text{km/h} $$

In these examples, it's clear that relative velocity not only depends on the speed of the moving objects but also on their direction of motion. Understanding relative velocity is essential for navigation, collision avoidance, and many other practical applications in physics and engineering.