Reference Frame Examples: 46 Problems with Answers

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

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

Concept Recap

A coordinate system attached to a particular observer that is used to describe the positions and motions of objects.

Are you 'moving' on a train? Depends on whether you ask someone on the train or the platform.

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: Reference Frame starts by naming what changes, over what time interval, and whether direction matters.

Common stuck point: Students often know a formula related to reference frame 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

easy

Train A moves east at

30 \text{ m/s}

and Train B moves east at

20 \text{ m/s}

, both relative to the ground. What is the velocity of Train A relative to Train B?

Answer

v_{A B} = 10 m/s east

First step

The velocity of A relative to B is:

v_{AB} = v_A - v_B

Full solution

2
$v_{A B} = 30 - 20 = 10 m/s east$
3
From the perspective of a passenger on Train B, Train A appears to move east at $10 \text{ m/s}$ .

A reference frame is a viewpoint from which motion is observed. Relative velocity between two objects is found by subtracting their velocities. Different observers can measure different velocities for the same object.

Example 2

medium

A boat travels at

5 \text{ m/s}

relative to the water, heading directly across a river that flows at

3 \text{ m/s}

. What is the boat's speed relative to the ground, and at what angle does it actually travel?

Example 3

medium

A boat moves at

5 \text{ m/s}

north relative to the water. The river flows east at

2 \text{ m/s}

. Find the boat's velocity relative to the ground (magnitude and direction).

Example 4

medium

In the ground frame, ball A moves east at

4 \text{ m/s}

and ball B moves north at

3 \text{ m/s}

. Find the speed of A in B's frame.

Example 5

medium

Walker A walks

1 \text{ m/s}

north on a ship moving

4 \text{ m/s}

east relative to the shore. Find A's velocity in the shore frame.

Example 6

hard

A pilot wants to fly due north. The wind blows

40 \text{ km/h}

east. Air speed of the plane is

250 \text{ km/h}

. What heading should the pilot choose, and what is the ground speed?

Example 7

hard

A river is

200 \text{ m}

wide; current is

1.5 \text{ m/s}

east. A boat aims directly across at

2 \text{ m/s}

relative to the water. How far downstream does it land?

Example 8

hard

A river is

80 \text{ m}

wide flowing east at

0.8 \text{ m/s}

. A boat can move at

2 \text{ m/s}

in still water. What heading angle (measured from due north) achieves zero downstream drift, and what is the crossing time?

Practice Problems

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

Example 1

medium

Car A travels north at

25 \text{ m/s}

and Car B travels south at

20 \text{ m/s}

, both relative to the ground. What is the velocity of Car A as seen by Car B?

Example 2

hard

A plane flies at

250 \text{ m/s}

due north relative to the air. A wind blows at

50 \text{ m/s}

from the west (toward the east). What is the plane's ground speed and actual direction of travel?

Example 3

easy

A passenger sits still in a moving train. What is her velocity in the train's reference frame?

Example 4

easy

The same passenger, viewed from the platform, when the train moves at

30

m/s. Her velocity?

Example 5

easy

Is there a single 'correct' inertial reference frame for measuring motion?

Example 6

easy

Two cars drive side by side at

25

m/s. What is one car's velocity relative to the other?

Example 7

easy

A ball dropped inside a smoothly moving train falls straight down in the train frame. Where does it land?

Example 8

easy

Why is 'the car moves at 60 km/h' an incomplete statement?

Example 9

easy

In the ground frame a thrown ball moves at

10

m/s east. In a frame moving east at

10

m/s, what is the ball's velocity?

Example 10

easy

Can an object be at rest in one frame and moving in another at the same time?

Example 11

medium

A person walks

2

m/s toward the front of a train moving

20

m/s. Find their velocity relative to the ground.

Example 12

medium

The same person walks

2

m/s toward the BACK of the

20

m/s train. Ground velocity?

Example 13

medium

Car A moves

30

m/s east, car B moves

20

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

Example 14

medium

Car A moves

30

m/s east, car B moves

20

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

Example 15

medium

A river flows

3

m/s east. A swimmer's velocity is

3

m/s east relative to water. Find the swimmer's velocity relative to the bank.

Example 16

medium

Inside an elevator accelerating upward, a scale reads more than your true weight. Which frame explains the extra reading simply: ground or elevator?

Example 17

challenge

Two trains approach on parallel tracks: A at

25

m/s east, B at

15

m/s west. A passenger on A sees B approach at what speed?

Example 18

challenge

A boat heads

4

m/s north across a river flowing

3

m/s east. Find the boat's speed relative to the bank.

Example 19

challenge

Standing on a train moving

10

m/s east, you throw a ball

5

m/s east (train frame) and it lands. In the ground frame, what is the ball's horizontal velocity?

Example 20

medium

A person walks

1.5

m/s toward the front of a bus moving

12

m/s. Find their ground velocity.

Example 21

medium

In a frame moving

5

m/s east, a ball measured at

3

m/s east. Find the ball's ground velocity.

Example 22

medium

Two boats: A at

6

m/s north, B at

6

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

Example 23

easy

A walker moves at

1.5 \text{ m/s}

east on a sidewalk that is fixed to the ground. In the ground frame, what is his velocity?

Example 24

easy

Cyclist A moves east at

8 \text{ m/s}

and cyclist B moves east at

5 \text{ m/s}

, both relative to the ground. What is A's velocity in B's frame?

Example 25

easy

An escalator moves up at

0.8 \text{ m/s}

. A man walks down the escalator at

0.8 \text{ m/s}

relative to the escalator. What is his velocity in the ground frame?

Example 26

medium

Two trains pass each other on parallel tracks: train A moves east at

30 \text{ m/s}

, train B moves west at

25 \text{ m/s}

. Find the speed of A as measured by an observer on B.

Example 27

medium

A passenger walks toward the front of a

20 \text{ m/s}

train at

1.5 \text{ m/s}

(in the train frame). Find her velocity in the ground frame.

Example 28

medium

A river flows east at

1 \text{ m/s}

. A boy swims east at

1.5 \text{ m/s}

relative to the ground. What is his velocity relative to the water?

Example 29

medium

Two cars approach an intersection: A moves east at

20 \text{ m/s}

, B moves north at

15 \text{ m/s}

. Find the speed of B as seen from A.

Example 30

medium

A coin is dropped inside a car moving east at

20 \text{ m/s}

at constant velocity. In the ground frame, what is the coin's horizontal velocity while falling?

Example 31

medium

In the ground frame a falling raindrop moves straight down at

5 \text{ m/s}

. A car moves east at

5 \text{ m/s}

. In the car frame, what angle do the streaks on the side window make with the vertical?

Example 32

medium

A boy throws a ball horizontally at

4 \text{ m/s}

from a flatbed truck moving at

10 \text{ m/s}

in the same direction. What is the ball's initial horizontal speed in the ground frame?

Example 33

hard

Plane A flies

200 \text{ km/h}

east, plane B flies

200 \text{ km/h}

60°

north of east. Find the speed of B in A's frame.

Example 34

hard

An observer in an accelerating bus (acceleration

a

forward) sees a ball on a frictionless tray. In the bus frame, what is the ball's apparent acceleration?

Example 35

hard

A ball is thrown straight up at

10 \text{ m/s}

from a flatbed moving

6 \text{ m/s}

east. In the ground frame, what is the initial velocity vector? Use

g

for falling.

Example 36

hard

An elevator accelerates upward at

2 \text{ m/s}^2

. In the elevator frame, what is the apparent gravitational acceleration? Use

g = 9.8 \text{ m/s}^2

Example 37

hard

Two cars start at the same point. A moves east at

20 \text{ m/s}

, B moves at

20 \text{ m/s}

90°

to A (north). After

5 \text{ s}

, how far apart are they?

Example 38

challenge

Two ships sail from the same port: A goes north at

15 \text{ km/h}

, B goes

60°

east of north at

20 \text{ km/h}

. At what rate does the distance between them increase?

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Related Concepts

Position Velocity Relative Velocity Inertia

Background Knowledge

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

positionvelocity