Kinematics and Relative Velocity

Understanding Relative Velocity

In physics, motion is described relative to a frame of reference. Whether you're on a moving train or a flying plane, your velocity depends on who is watching.

We use subscripts to keep track:

P/G: Object relative to Ground
P/A: Object relative to Medium (Air/Water)
A/G: Medium relative to Ground

This is the wind velocity. It represents the movement of the air itself relative to the stationary ground. This is the airspeed. It's how fast the plane's engines are pushing it through the surrounding air molecules. This is the ground speed. It is the actual path the plane travels over the earth, combining the plane's effort and the wind's push. Welcome! In the real world, motion is rarely seen from a perfectly stationary point. Imagine this plane. Its speed through the air is one thing, but its speed relative to the person on the ground is another. We use specific subscripts to track these relationships: P for Plane, A for Air, and G for Ground.

Velocity is always relative to a frame of reference.
Subscript notation clarifies the relationship between moving objects.
The ground is usually our stationary reference point.

The Vector Chain Rule

The relationship between these velocities is governed by the Chain Rule of vector addition:

v_P/G = v_P/A + v_A/G

Notice how the 'A' subscripts 'cancel out' in the middle, leaving you with P and G.

To find the resultant velocity, we use the vector chain rule. Notice the pattern: the inner subscripts must match. If we add the velocity of the plane relative to the air to the velocity of the air relative to the ground, we get the plane's velocity relative to the ground. Geometrically, this is just tip-to-tail addition.

Relative velocities add as vectors, not scalars.
The 'Chain Rule' helps organize the addition: (Object/Medium) + (Medium/Ground) = (Object/Ground).
Tip-to-tail addition is the standard geometric approach.

2D Kinematics & Acceleration

In 2D motion, we analyze horizontal ($x$) and vertical ($y$) components independently.

Average Acceleration:
a_avg = (v_f - v_i) / Δt

Remember: Acceleration occurs if direction changes, even if speed stays constant!

Moving in two dimensions means tracking changes in both X and Y. Look at this particle moving in a curve. Even if its speed is constant, the direction of its velocity vector is changing. Because the vector is changing, there is a non-zero acceleration vector. We calculate this by subtracting the initial velocity from the final velocity.

Horizontal and vertical motions do not affect each other.
Acceleration is a change in the velocity vector (magnitude OR direction).
Average velocity is displacement divided by time.

Interactive Scenario: The Airplane Crosswind

A pilot wants to fly due North, but a wind is blowing from the West (to the East). Use the slider to adjust the plane's 'heading' to compensate.

Let's put this into practice. You need to reach the airport due North. But look—the wind is pushing you East. Try adjusting the plane's heading. You'll need to 'crab' into the wind to keep your ground track straight. Perfect! By pointing North-West, your resultant motion is now purely Northward. This is how pilots navigate crosswinds every day. Notice how the resultant vector changes as you rotate the plane. You are looking for the angle where the Eastward wind is perfectly cancelled by your Westward heading component.

To travel straight, you must aim into the wind.
The 'heading' is the direction the plane points (v_P/A).
The 'track' is the actual path over ground (v_P/G).

Step-by-Step Problem Solving

Don't just guess! Follow this workflow for every 2D vector problem:

Sketch: Draw vectors tip-to-tail.
Components: Use sin/cos to find x and y.
Equations: Solve X and Y separately.
Combine: Use Pythagoras for the final magnitude.

To solve these problems accurately, follow these four steps. First, sketch the vectors. Identify your resultant—usually the motion relative to the ground. Second, break every vector into horizontal and vertical components. Third, write your equations for X and Y separately. Finally, use the Pythagorean theorem to find the final speed if needed. Never just add the speeds together!

Always sketch the vector diagram first.
Break vectors into components before adding.
Never add magnitudes directly unless parallel.

Writing Exercise: The Pitfall Check

A student says: 'If my plane flies at 200 km/h and there is a 50 km/h crosswind, my ground speed is always 250 km/h.'

Explain why this is incorrect and what they should do instead.

One of the most common mistakes is adding magnitudes directly. Look at the student's statement. Why is it wrong? Type your explanation and submit it for feedback.

Magnitudes do not add linearly in 2D.
Vector addition requires components.
Direction matters.