Relative Velocity Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Relative Velocity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Relative Velocity starts by naming what changes, over what time interval, and whether direction matters.

Common stuck point: 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.

Sense of Study hint: Ask: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?

Worked Examples

Example 1

medium

A plane has airspeed

200\text{ m/s}

east. A

50\text{ m/s}

wind blows from north to south. Find ground velocity (magnitude and direction).

Plane 200 m/s east + wind 50 m/s south → ground velocity ≈ 206 m/s at 14° south of east

Answer

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

First step

v_x = 200

m/s east,

v_y = -50

m/s (south).

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Example 2

hard

A boat must cross a

100\text{ m}

-wide river to a point directly opposite. Boat speed relative to water is

5\text{ m/s}

; current is

3\text{ m/s}

. At what angle upstream must the bow point, and what is the crossing time?

Boat upstream heading + current → resultant due north; crossing time = 25 s

Example 3

hard

A boat heads

8\text{ m/s}

upstream against a

3\text{ m/s}

current. After

60\text{ s}

it turns and heads downstream at

8\text{ m/s}

relative to water. Find ground displacement after a total of

120\text{ s}

Example 4

challenge

From a car moving

25\text{ m/s}

east, a passenger throws a ball

15\text{ m/s}

east relative to the car. A pedestrian on the ground observes the ball. Find the ball's ground velocity. If the car then decelerates at

2\text{ m/s}^2

, does the ball's ground velocity change after release?

Car 25 m/s east + ball thrown 15 m/s east (relative to car) → ball ground velocity 40 m/s east

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Car A moves

20

m/s east, car B moves

12

m/s east. Find A's velocity relative to B.

Example 2

easy

Two cars approach head-on: A at

30

m/s east, B at

20

m/s west. Find their closing speed.

Example 3

easy

A boat moves

5

m/s east relative to water; the water moves

2

m/s east. Boat speed relative to ground?

Example 4

easy

From a passenger's frame in a

25

m/s train, the trees appear to move how?

Example 5

easy

Two runners side by side at

4

m/s. Velocity of one relative to the other?

Example 6

easy

A walkway moves

1

m/s; a person walks

1.5

m/s on it in the same direction. Speed relative to ground?

Example 7

easy

Why must direction be tracked when combining velocities?

Example 8

easy

A jet flies

200

m/s east; a tailwind blows

30

m/s east. Ground speed?

Example 9

medium

A jet flies

200

m/s east into a

30

m/s east headwind... actually the wind blows west at

30

m/s. Find the ground speed.

Example 10

medium

A boat heads

6

m/s north across a river flowing

8

m/s east. Find its ground speed.

Boat heading north + eastward river current → resultant ground velocity

Example 11

medium

Train A at

40

m/s east passes train B at

25

m/s east on a parallel track. How fast does B appear to move in A's frame?

Example 12

medium

A swimmer crosses a river by heading

3

m/s relative to water; current is

4

m/s. Find the swimmer's ground speed.

Swimmer velocity + river current → resultant 5 m/s

Example 13

medium

Person walks

2

m/s forward on a train going

18

m/s; another walks

2

m/s backward. Find each person's ground velocity.

Example 14

medium

Two cyclists ride toward each other, each at

5

m/s. A fly starts on one and flies

10

m/s to the other and back repeatedly until they meet

20

s later. Total fly distance?

Example 15

challenge

Two trains

1000

m apart approach: A at

30

m/s, B at

20

m/s (toward each other). When do they meet?

Example 16

challenge

A boat must cross a

120

m wide river straight across; it swims

5

m/s relative to water while the current is

3

m/s. At what angle upstream must it head, and what is the crossing time?

Boat heading upstream + current → resultant straight across (north)

Example 17

challenge

In still air a plane flies

250

m/s. It must fly due north while a

70

m/s east wind blows. Find its ground speed due north (round to nearest m/s).

Plane heading + eastward wind → resultant due north at 240 m/s

Example 18

medium

Car A at

35

m/s east, car B at

20

m/s east. Find A's velocity relative to B.

Example 19

medium

A boat heads

8

m/s north across a river flowing

6

m/s east. Find its ground speed.

Boat 8 m/s north + river 6 m/s east → ground speed 10 m/s

Example 20

medium

Two cars approach head-on, A at

22

m/s and B at

18

m/s. Find their closing speed.

Example 21

easy

Car A moves

15\text{ m/s}

east; car B moves

10\text{ m/s}

east. Find A's velocity relative to B.

Example 22

easy

Two cars approach head-on: A at

25\text{ m/s}

east, B at

15\text{ m/s}

west. Find their closing speed.

Example 23

easy

A boat moves

4\text{ m/s}

east relative to water; water moves

3\text{ m/s}

west. Find the boat's speed relative to ground.

Example 24

easy

Two runners both jog north at

3\text{ m/s}

. Velocity of one relative to the other?

Example 25

easy

A walkway moves

1.2\text{ m/s}

east. A person walks

0.8\text{ m/s}

east on it. Speed relative to ground?

Example 26

easy

A walkway moves

1.2\text{ m/s}

east. A person walks

0.8\text{ m/s}

west on it. Speed relative to ground?

Example 27

medium

A boat heads

3\text{ m/s}

north across a river flowing

4\text{ m/s}

east. Find ground speed.

Boat 3 m/s north + river 4 m/s east → ground speed 5 m/s

Example 28

medium

A jet flies

180\text{ m/s}

east into a

30\text{ m/s}

west headwind. Find ground speed.

Example 29

medium

Train A at

25\text{ m/s}

east passes train B at

15\text{ m/s}

east on a parallel track. Velocity of B in A's frame?

Example 30

medium

A swimmer swims

2\text{ m/s}

relative to water across a current of

1.5\text{ m/s}

at right angles. Find swimmer's ground speed.

Swimmer 2 m/s + current 1.5 m/s → ground speed 2.5 m/s

Example 31

medium

A car drives

20\text{ m/s}

east. Rain falls vertically at

10\text{ m/s}

in the ground frame. Find the rain's velocity in the car's frame.

Example 32

medium

A passenger walks

1\text{ m/s}

forward on a

20\text{ m/s}

train. A second passenger walks

1\text{ m/s}

backward. Find each passenger's ground velocity.

Example 33

medium

A boat heads

12\text{ m/s}

north across a

5\text{ m/s}

east river current. Find ground speed.

Boat 12 m/s north + river 5 m/s east → ground speed 13 m/s

Example 34

medium

Two cars on a highway, A at

30\text{ m/s}

east, B at

25\text{ m/s}

east. How long does A take to close a

200\text{ m}

gap behind B?

Example 35

medium

Two cars approach head-on: A at

18\text{ m/s}

east, B at

12\text{ m/s}

west, initially

300\text{ m}

apart. When do they meet?

Example 36

medium

A swimmer moves

1.5\text{ m/s}

relative to water; the river current is

2.0\text{ m/s}

. Swimmer heads directly across (perpendicular to current). Find ground speed.

Swimmer 1.5 m/s + current 2.0 m/s → ground speed 2.5 m/s

Example 37

medium

A plane flies

150\text{ m/s}

east with a

40\text{ m/s}

east tailwind. Find its ground speed.

Example 38

hard

A river is

80\text{ m}

wide; current is

3\text{ m/s}

. A boat travels

5\text{ m/s}

relative to water aimed directly across. Find crossing time and downstream drift.

Boat 5 m/s across + current 3 m/s → resultant northeast, drift

Example 39

hard

In still air a plane flies

300\text{ m/s}

. It must fly due east while a

60\text{ m/s}

wind blows from the south. Find its ground speed due east.

Example 40

hard

Two trains

600\text{ m}

apart approach: A at

20\text{ m/s}

, B at

10\text{ m/s}

(toward each other). When do they meet?

Example 41

hard

Two cyclists ride toward each other, each at

4\text{ m/s}

, initially

200\text{ m}

apart. A bee flies

8\text{ m/s}

between them until they meet. Find total bee distance.

Example 42

hard

A passenger inside a train walks

1.5\text{ m/s}

forward. The train accelerates from

10\text{ m/s}

20\text{ m/s}

over

10\text{ s}

. Find the passenger's ground velocity at

t = 10\text{ s}

assuming her walking speed is constant in the train frame.

Example 43

challenge

A plane has airspeed

250\text{ m/s}

and must fly directly north while a

50\text{ m/s}

wind blows from the east. Find the heading angle (measured from north) and the ground speed north.

Plane heading east of north + westward wind → resultant due north at ≈245 m/s

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Related Concepts

Reference Frame Velocity Vectors

Background Knowledge

These ideas may be useful before you work through the harder examples.

reference framevelocityvectors