Relative Velocity Formula

Relative velocity is the velocity of one object as measured from the reference frame of another object.

The Formula

v⃗A/C=v⃗A/B+v⃗B/C\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}

When to use: How fast something seems to move depends on who is watching.

Quick Example

If you walk forward at 2 m/s inside a train moving at 20 m/s relative to the ground, your velocity relative to the ground is about 22 m/s forward.

Notation

v⃗A/B\vec{v}_{A/B} means 'velocity of A relative to B'.

What This Formula Means

Relative velocity is the velocity of one object as measured from the reference frame of another object.

How fast something seems to move depends on who is watching.

Formal View

In Galilean relativity, relative velocities add vectorially: v⃗A/C=v⃗A/B+v⃗B/C\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}.

Worked Examples

Example 1

medium
A plane has airspeed 200 m/s200\text{ m/s} east. A 50 m/s50\text{ m/s} wind blows from north to south. Find ground velocity (magnitude and direction).

Answer

≈206 m/s, 14.0∘ south of east\approx 206 \text{ m/s, } 14.0^\circ \text{ south of east}

First step

1
vx=200v_x = 200 m/s east, vy=−50v_y = -50 m/s (south).

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Example 2

hard
A boat must cross a 100 m100\text{ m}-wide river to a point directly opposite. Boat speed relative to water is 5 m/s5\text{ m/s}; current is 3 m/s3\text{ m/s}. At what angle upstream must the bow point, and what is the crossing time?

Example 3

hard
A boat heads 8 m/s8\text{ m/s} upstream against a 3 m/s3\text{ m/s} current. After 60 s60\text{ s} it turns and heads downstream at 8 m/s8\text{ m/s} relative to water. Find ground displacement after a total of 120 s120\text{ s}.

Common Mistakes

  • Mixing velocities measured in different frames without converting them. - Fix this by naming the system, checking "Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?", and attaching units or direction to the final statement.
  • Ignoring direction when combining velocities. - Fix this by naming the system, checking "Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?", and attaching units or direction to the final statement.
  • Using relative velocity from a keyword alone - Signal words like position, speed, velocity only point to a possible model; the system must match too.
  • Substituting numbers before defining the system - A formula cannot repair a missing object, boundary, direction, medium, or circuit path.

Why This Formula Matters

Relative Velocity helps students describe motion precisely instead of relying on everyday words like fast or slow. It prepares them to interpret graphs, choose equations, and connect motion to forces and energy.

Frequently Asked Questions

What is the Relative Velocity formula?

Relative velocity is the velocity of one object as measured from the reference frame of another object.

How do you use the Relative Velocity formula?

How fast something seems to move depends on who is watching.

What do the symbols mean in the Relative Velocity formula?

v⃗A/B\vec{v}_{A/B} means 'velocity of A relative to B'.

Why is the Relative Velocity formula important in Physics?

Relative Velocity helps students describe motion precisely instead of relying on everyday words like fast or slow. It prepares them to interpret graphs, choose equations, and connect motion to forces and energy.

What do students get wrong about Relative Velocity?

Students often know a formula related to relative velocity but skip the recognition step: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated? That leads to a correct-looking substitution attached to the wrong physical model.

What should I learn before the Relative Velocity formula?

Before studying the Relative Velocity formula, you should understand: reference frame, velocity, vectors.

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