relative motion

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An important concert in the study of physics, and in particular, the study of mechanics, is the concept that there is no absolute frame of reference. What this means can be expressed in two ways.
First, when any two observers measure the motion of an object, what they measure is determined by the observer.
For example, watch the video of the swan below.

The swan appears to be stationary when the camera is stationary. If the camera is moving, this one can either appear moving forward or moving backward, depending on what the camera is doing. All are correct as they depend on what the camera sees. What we therefore say is that the motion of the swan is relative to the camera in this case.

Another way we can express this is that there is no absolute frame of reference, or what we determined to be the true absolute origin - the coordinates being (0, 0) in a 2 dimensional space, or (0, 0, 0) in a three dimensional space.
These coordinates are the origin origin for the OBSERVER, with each observer determining what they see as their origin

One of the most foundational principles in physics and classical mechanics is that all motion is relative—there is no single, fixed, or absolute frame of reference in the universe.

Whenever we describe an object's speed, direction, or position, our measurements depend entirely on the perspective of the observer. For instance, if you are sitting on a high-speed train, you appear completely stationary to the passenger next to you. However, to an observer standing on the platform outside, you are moving at over 100 km/h. Both observers are correct within their own coordinate systems.

Frames of Reference & Vector Addition

To analyze relative motion mathematically, we define a frame of reference—a coordinate system attached to a chosen observer, where they place their origin (0, 0, 0):

1D Relative Velocity: For motion along a single straight line, the velocity of Object A relative to Observer B (v_A/B) is the vector difference between their individual velocities relative to the ground (v_A and v_B):

v_A/B = v_A - v_B

2D Relative Velocity: When objects move across two dimensions (such as a boat crossing a flowing river or an aircraft flying in a crosswind), relative velocities are calculated using vector addition:

v_object/bank = v_object/fluid + v_fluid/bank

Watch the video below to explore relative motion in one and two dimensions, and see how vector addition lets us translate motion across different frames of reference.

Sample Problem
We are now ready to try a sample problem
Below is a sample problem with a video that explain how to solve it. It is suggested you try the problem beforehand, as this actually aids understanding, even if you are unsure if you are correct.

Interactive

Let's apply what you've learnt.

The following interactive (courtesy Tom Wash) allows you to set the river velocity and the velocity of the boat relative to the river. It then determines the velocity of the boat relative to the Earth.

Are useful way to use is to set the variables and then calculate the velocity of the boat, using the animation to check your answer.

Practice Problems: Relative Motion & Vectors

1. An aeroplane has a velocity of 240 km/h north according to its instruments. A crosswind of 100 km/h is blowing from the west. Where will the aircraft be after two hours?

Resultant Velocity Calculation:

Since the plane heads North and the wind blows from the west (toward the East), the components are perpendicular:

v_resultant = √(240² + 100²) = √(57600 + 10000) = √67600 = 260 km/h

Direction: tan(θ) = 100 / 240 → θ = tan^-1(0.4167) ≈ 22.6° East of North (N23°E)

Displacement after 2 hours:

Distance = 260 km/h × 2 h = 520 km

Answer: 520 km, N23°E (or 23° East of North)

2. A boat travels at 10 m/s North relative to the water in a river where the current is 10 m/s West. Find the velocity of the boat relative to the riverbank.

Vector Addition:

Magnitude: v = √(10² + 10²) = √200 ≈ 14.14 m/s

Direction: Since North and West components are equal in magnitude (10 m/s each), the direction is exactly Northwest (NW, or N45°W).

Answer: 14 m/s, NW

3. A sailor stands on the deck of a ship traveling at 20 km/h north. He looks up and sees an albatross flying at 28 km/h southeast relative to the ship. What is the velocity of the albatross relative to the ocean floor?

Relative Velocity Formula:

v_bird/ocean = v_bird/ship + v_ship/ocean

Break into components:

• v_ship/ocean = 0 km/h East, 20 km/h North

• v_bird/ship (28 km/h SE at 45°):

East component = 28 × cos(45°) ≈ 19.8 km/h East
North component = -28 × sin(45°) ≈ -19.8 km/h North (19.8 km/h South)

Summing components:

• East total = 0 + 19.8 = 19.8 km/h East ≈ 20 km/h East
• North total = 20 - 19.8 = 0.2 km/h North ≈ 0 km/h

Answer: 20 km/h East

4. Three swimmers can swim equally fast relative to the water and hold a race across a river:
• Swimmer A swims perpendicular to the current and lands downstream on the far bank.
• Swimmer B swims upstream at an angle so that he lands directly opposite the starting point.
• Swimmer C swims downstream at an angle to take advantage of the current.

Who crosses the river in the least time? Account for your answer.

Explanation:

Crossing time depends only on the velocity component perpendicular to the riverbanks (the across-river direction):

t = width / v_{perpendicular}

• Swimmer A directs 100% of their swimming speed straight across the river, maximizing v_{perpendicular}.

• Swimmers B and C angle their heading, meaning a portion of their swimming speed is directed upstream or downstream, leaving a smaller perpendicular component across the river.

Answer: Swimmer A crosses in the least time because all of their effort is directed straight across the river.

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Frames of Reference & Vector Addition

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