Skip to content

Relative Motion

Source lecture(s): SC133 Lec 5

Intuition

Velocity is always relative to something. Walk forward at \(1\,\text{m/s}\) inside a train doing \(30\,\text{m/s}\): the ground sees you at \(31\,\text{m/s}\); a fellow passenger sees \(1\,\text{m/s}\); you see yourself at rest. None of these is "the real" velocity — each is correct in its own reference frame. Physics only demands that observers agree on the laws, not on the velocities.

The velocity-addition rule

Label frames and objects with subscripts — "P relative to A" as \(\vec v_{PA}\). The chain rule of frames:

\[\boxed{\,\vec v_{PA} = \vec v_{PB} + \vec v_{BA}\,}\]

Read the subscripts like dominoes: \(PA = PB + BA\) (inner letters match and cancel). Reversing a pair flips the sign: \(\vec v_{AB} = -\vec v_{BA}\).

Because the rule is a vector equation, it works in 2-D: this is where vector components earn their keep.

Worked example: crossing a river (two strategies)

River flows east at \(3\,\text{m/s}\); boat does \(5\,\text{m/s}\) relative to the water.

Fastest crossing: aim straight across. Ground velocity = \(5\) across + \(3\) downstream = \(\sqrt{34} \approx 5.8\,\text{m/s}\); you land downstream but in minimum time.

Straight-line crossing: aim upstream at angle \(\sin\theta = 3/5 \Rightarrow \theta = 37°\). Ground speed across: \(\sqrt{5^2 - 3^2} = 4\,\text{m/s}\). You land directly opposite, but the crossing takes longer.

Same physics, different goals — the classic exam pair.

Worked example: rain on a moving car

Rain falls vertically at \(8\,\text{m/s}\); you drive at \(6\,\text{m/s}\). In your frame the rain has a horizontal component \(6\,\text{m/s}\) toward you: it appears to fall at \(\tan^{-1}(6/8) \approx 37°\) from vertical, at \(10\,\text{m/s}\). This is why the front windscreen catches rain and the rear window stays dry.

Frames and the laws of physics

All frames moving at constant velocity (inertial frames) agree on accelerations — \(\vec a_{PA} = \vec a_{PB}\) when \(\vec v_{BA}\) is constant — so they agree on Newton's laws. Accelerating frames disagree, and require fictitious forces; that thread leads to the rotating-bucket problem in fluids and beyond.

Common mistakes

  • Adding speeds instead of velocities. Direction matters — use components.
  • Scrambling subscripts. Write the chain out (\(v_{PA} = v_{PB} + v_{BA}\)) and check the inner letters cancel before plugging numbers.
  • Believing one frame is "really" at rest. The ground is just a convenient frame — it's also on a spinning planet orbiting a star.

Knowledge graph position

Prerequisites: Vectors, Kinematics. Leads to: Newton's laws (inertial frames), collision analysis in the center-of-mass frame.

Quiz

Q1 (computational). Plane airspeed \(200\,\text{km/h}\) pointing north; wind \(50\,\text{km/h}\) blowing east. Ground speed and track?

Answer

\(v = \sqrt{200^2 + 50^2} \approx 206\,\text{km/h}\), heading \(\tan^{-1}(50/200) \approx 14°\) east of north.

Q2 (conceptual). Two cars approach each other, each at \(60\,\text{km/h}\). What is the velocity of car A in car B's frame?

Answer

\(120\,\text{km/h}\) toward B: \(\vec v_{AB} = \vec v_{A,\text{gnd}} - \vec v_{B,\text{gnd}} = 60 - (-60) = 120\) km/h.

Q3 (multiple choice). You drop a ball inside a train moving at constant velocity. It lands: (a) behind your hand (b) directly below your hand (c) ahead of your hand

Answer

(b). The ball keeps the train's horizontal velocity (Newton's first law); in the train frame the physics is identical to standing still — the definition of an inertial frame.