5bRelative Motion and Frames of Reference
Discover how motion depends on the observer's reference frame and how to add velocity vectors for boats, planes, and moving observers.
Understanding relative motion is essential for navigation, aviation, and any situation where multiple objects move simultaneously — it is also the conceptual gateway to Einstein's special relativity.
Lesson Overview
Motion is always measured relative to a chosen reference frame. Galilean relativity tells us that the laws of mechanics are the same in all inertial (non-accelerating) frames. The relative velocity of object A with respect to observer B is found by vector subtraction: vAB = vA − vB. Classic examples include a boat crossing a river and an airplane flying in wind.
Key Concepts
Reference Frame
A coordinate system used to describe the position and motion of objects.
Relative Velocity
v_AB = v_A − v_B; the velocity of A as measured by observer B.
Inertial Frame
A non-accelerating reference frame where Newton's laws hold without modification.
Galilean Relativity
The laws of mechanics are identical in all inertial frames; only relative motion is measurable.
Boat-River Problem
The boat's velocity relative to the ground = boat velocity relative to water + water velocity relative to ground.
Vector Addition
Relative velocities are added/subtracted as vectors, requiring both magnitude and direction.
A train moves east at 60 m/s. A passenger walks east at 2 m/s relative to the train. What is the passenger's velocity relative to the ground?
Using the same train, another passenger walks west at 2 m/s relative to the train. What is their velocity relative to the ground?
A boat can travel at 4 m/s in still water. A river flows east at 3 m/s. The boat heads north. Find the boat's speed and direction relative to the ground.
An airplane flies north at 250 m/s relative to the air. The wind blows east at 50 m/s. Find the plane's velocity relative to the ground.
Car A moves east at 30 m/s. Car B moves west at 20 m/s. What is the velocity of A relative to B?
A swimmer can swim at 1.5 m/s in still water. A river flows at 1.0 m/s. The swimmer heads straight across. What is the swimmer's speed relative to the ground?
Hint: The two velocity vectors are perpendicular. Use the Pythagorean theorem.
In the swimmer problem above, if the river is 60 m wide, how far downstream does the swimmer land?
Hint: Find the time to cross using the swimmer's speed perpendicular to the river, then find how far the current carries them.
A plane needs to fly due north. The wind blows east at 80 m/s. The plane's airspeed is 200 m/s. At what angle west of north must the pilot aim?
Hint: The plane must aim into the wind so that the resultant velocity points north. Draw the vector triangle.
Two cars approach each other on a highway. Car A goes 25 m/s north; Car B goes 30 m/s south. What is the velocity of A relative to B?
Hint: Use v_AB = v_A − v_B. Be careful with signs — choose a positive direction first.
Why is it impossible to define 'absolute rest' according to Galilean relativity?
Hint: Think about what Galilean relativity says about the laws of physics in different inertial frames.
Key Vocabulary
Reference Frame
A coordinate system attached to an observer, used to measure position, velocity, and acceleration.
Example: A person on a train and a person on the platform use different reference frames.
Relative Velocity
The velocity of one object as measured by an observer in a particular reference frame.
Example: v_AB = v_A − v_B gives the velocity of A relative to B.
Inertial Frame
A reference frame that is not accelerating; Newton's laws hold without modification in inertial frames.
Example: A train moving at constant velocity is an inertial frame; a braking train is not.
Galilean Relativity
The principle that the laws of mechanics are the same in all inertial reference frames.
Example: A ball dropped inside a smoothly moving train falls straight down to a passenger inside, just as it would at rest.
Interactive Practice — 5 Questions
A car moves east at 40 m/s. A truck moves east at 25 m/s. What is the velocity of the car relative to the truck?
A boat heads north at 5 m/s; the river flows east at 5 m/s. What is the boat's speed relative to the ground?
According to Galilean relativity, which of the following is the same in all inertial frames?
An airplane flies at 300 m/s relative to the air. A tailwind blows at 50 m/s in the same direction. What is the plane's ground speed?
Two trains approach each other at 20 m/s each. What is the velocity of train A relative to train B?
Independent Practice
A boat travels at 6 m/s relative to the water. The river current is 4 m/s east. The boat heads north. Find (a) the boat's speed relative to the ground and (b) its direction of travel.
A passenger on a train moving at 30 m/s east throws a ball at 10 m/s west relative to themselves. What is the ball's velocity relative to the ground?
An airplane must fly due east. The wind blows south at 60 m/s. The plane's airspeed is 180 m/s. Find the heading angle and the ground speed.
Explain in your own words why a person standing still on Earth is not in a truly "absolute" rest frame.
★ A river is 120 m wide and flows south at 2 m/s. A swimmer can swim at 3 m/s in still water. At what angle upstream must the swimmer aim to land directly across from the starting point? How long does the crossing take?
ChallengeCommon Mistakes
Adding speeds as scalars when velocities point in different directions.
Relative velocities are vectors. Use vector addition/subtraction with components or the Pythagorean theorem for perpendicular cases.
Confusing v_AB with v_BA — thinking they are the same.
v_AB = −v_BA. The velocity of A relative to B is equal and opposite to the velocity of B relative to A.
Forgetting to account for wind or current when finding ground speed.
Ground velocity = velocity relative to medium + velocity of medium relative to ground.
Math Tips
Use subscript notation carefully: v_AB means "velocity of A as seen by B." The formula is v_AB = v_A(ground) − v_B(ground). Always define a positive direction before plugging in numbers.
For perpendicular vectors (e.g., boat crossing river), use the Pythagorean theorem for magnitude and arctan for direction. For non-perpendicular cases, break into x and y components.