3.5 Addition of Velocities

Relative Velocity and Frames of Reference

Relative velocity describes how fast an object moves from the perspective of a particular observer. Since different observers can be moving differently, the same object can have different velocities depending on who's watching. This idea sits at the heart of two-dimensional kinematics problems involving boats, planes, and anything else moving through a moving medium.

Vector Addition for Relative Velocity

Velocity is a vector, so finding relative velocity means doing vector math, not just subtracting speeds. The core equation is:

$\vec{v}_{rel} = \vec{v}_{obj} - \vec{v}_{ref}$

where $\vec{v}_{obj}$ is the object's velocity and $\vec{v}_{ref}$ is the velocity of the reference frame you're observing from.

In two dimensions, you can carry out this subtraction two ways:

• Component method: Break each velocity into x- and y-components, then subtract corresponding components separately.
• Head-to-tail method: Draw $\vec{v}_{obj}$, then draw $-\vec{v}_{ref}$ (same magnitude, flipped direction) starting from the tip of $\vec{v}_{obj}$. The resultant arrow from the tail of the first to the tip of the second is $\vec{v}_{rel}$.

Quick examples to build intuition:

• A car moving at 60 km/h and a train moving at 80 km/h in the same direction: from the train's frame, the car moves at $60 - 80 = -20$ km/h (it appears to drift backward at 20 km/h).
• A boat sailing at 10 knots into a 2-knot current flowing against it: relative to the ground, the boat moves at $10 - 2 = 8$ knots.

Frame of Reference in Velocity Measurements

A frame of reference is the coordinate system an observer uses to measure position and velocity. The velocity you measure for an object depends entirely on which frame you're in.

• A passenger sitting on a moving train has zero velocity in the train's frame, but might be traveling at 90 km/h in the Earth's frame.
• A satellite orbiting Earth is moving at roughly 7.8 km/s relative to Earth's surface, but has a completely different velocity relative to the Sun.

Galilean relativity tells us that the laws of physics work the same way in all inertial frames of reference (frames moving at constant velocity, with no acceleration). When you switch between inertial frames, you just add or subtract the relative velocity between the frames. This process is called a reference frame transformation, and it's what the equation $\vec{v}_{rel} = \vec{v}_{obj} - \vec{v}_{ref}$ captures.

Applications of Relative Velocity

Motion Across Different Mediums

When an object moves through a medium that is itself moving (a river current, a crosswind), the object's velocity relative to the ground is the vector sum of two velocities: its velocity relative to the medium, and the medium's velocity relative to the ground.

How to solve these problems:

1. Identify the velocity vectors. For a boat in a river, that's the boat's velocity relative to the water and the water's velocity relative to the shore.
2. Set up a coordinate system. Typically, one axis points across the medium (e.g., across the river) and the other points along it (e.g., downstream).
3. Decompose into components if the vectors aren't aligned with your axes.
4. Add the components to get the resultant velocity relative to the ground.
5. Find magnitude and direction of the resultant using the Pythagorean theorem and inverse tangent.

For a plane flying in a crosswind, the same logic applies. If the pilot wants to travel due north but there's a wind blowing east, the pilot has to aim slightly west of north so the wind-corrected path ends up pointing where they want to go. The ground velocity (what an observer on the ground sees) is the vector sum of the airspeed vector and the wind vector.

Velocity Composition in Classical Mechanics

The Galilean transformation is the formal name for the rule that velocities add linearly between inertial frames. If frame B moves at velocity $\vec{v}_{B}$ relative to frame A, and an object moves at $\vec{v}_{obj,B}$ in frame B, then its velocity in frame A is:

$\vec{v}_{obj,A} = \vec{v}_{obj,B} + \vec{v}_{B}$

This linear addition works for all everyday speeds. It only breaks down when objects approach the speed of light, which is well outside the scope of introductory mechanics. For every problem you'll encounter in this course, Galilean velocity addition is the right tool.