F. Revolutions in Modern Physics: Quantum Mechanics and Special Relativity · Special RelativityFrames of Reference and Galilean Relativity
F1.1 F2.1About 65 minutes
Learning goals- I can describe a frame of reference and tell inertial frames from non-inertial frames.
- I can use the Galilean transformation to change positions and velocities from one inertial frame to another.
- I can state Galileo's principle of relativity and explain what it means for experiments inside a moving vehicle.
- I can explain why the speed of light predicted by Maxwell's equations created a problem for Newtonian physics.
Words to know
frame of reference
A coordinate system plus a set of synchronized clocks, used by an observer to measure positions and times of events.
event
Something that happens at one place at one instant, described by a position (x, y, z) and a time t.
inertial frame
A frame of reference that is not accelerating, in which Newton's first law holds: an object with no net force stays at rest or moves at constant velocity.
non-inertial frame
An accelerating (or rotating) frame of reference, in which objects seem to be pushed by fictitious forces.
Galilean transformation
The 'common sense' equations x = x′ + vt and t = t′ that convert measurements between two inertial frames moving at relative velocity v.
principle of relativity (Galilean)
The laws of mechanics are the same in every inertial frame, so no mechanical experiment can detect uniform motion.
absolute time
Newton's assumption that time passes at the same rate for every observer, no matter how they move.
Who is really moving?
You are on a GO train, reading. Your water bottle sits on the tray in front of you. To you, the bottle is at rest. To a friend waiting on the platform, the same bottle is moving at 100 km/h. Who is right?
Both. Motion is always measured relative to something. Physics makes this precise with the idea of a frame of reference. This whole unit is about what happens when observers in different frames compare their measurements. At everyday speeds, the answers match common sense. Near the speed of light, they do not, and that discovery changed physics forever.
Frames of reference and events
A frame of reference is a coordinate system (axes and a ruler) together with clocks at every point, all synchronized. An observer uses it to record events. An event is something that happens at one place at one instant: a firecracker bursting, a ball landing, a flash of light. In one frame an event has coordinates (x, y, z, t).
Remember from Dynamics: in the unit Dynamics: Forces and Motion in a Plane you met inertial and non-inertial frames.
- An inertial frame is not accelerating. Newton's first law works in it. A train moving at constant velocity on a straight, smooth track is (very nearly) inertial.
- A non-inertial frame accelerates. A car that is braking, or a merry-go-round that is turning, is non-inertial. In it, objects seem to be pushed by fictitious forces (you lurch forward when the bus brakes, though nothing pushed you).
Special relativity deals only with inertial frames. (Accelerating frames and gravity are handled by Einstein's later theory, general relativity.)
A frame at the surface of Earth is not perfectly inertial, because Earth spins and orbits the Sun, but the accelerations are small enough that we can treat the ground as inertial for most problems.
Galileo's ship
In 1632, Galileo Galilei described a thought experiment. Shut yourself in a cabin below the deck of a large ship, with butterflies, a fish bowl and dripping water. When the ship sails smoothly at constant speed, everything behaves exactly as it does when the ship is tied up at the dock: the drops fall straight down, the fish swim equally easily in every direction, and a ball tossed to a friend takes the same effort either way. You cannot tell whether the ship is moving.
This is the Galilean principle of relativity:
The laws of mechanics are the same in every inertial frame. No mechanical experiment done inside a closed, smoothly moving frame can detect its uniform motion.
There is no experiment that says "this frame is truly at rest." Only relative motion can be measured.
The Galilean transformation
Let frame S be the ground, and frame S′ be a train moving in the +x direction at constant velocity v relative to S. Start both clocks at t = 0 when the origins O and O′ are at the same place. Newton assumed that time is absolute: both observers' clocks always agree. Then, for any event:
- x = x′ + vt (position)
- y = y′, z = z′ (no motion across the track)
- t = t′ (time is the same for everyone)
Dividing changes in position by the (shared) time gives the Galilean velocity addition rule:
u = u′ + v
where u′ is an object's velocity in S′ and u is its velocity in S. You used this in the Grade 11 unit Kinematics: Motion in One and Two Dimensions for boats on rivers and planes in wind. Velocities are vectors, so take direction into account: a ball rolled toward the back of the train has u′ negative.
Worked exampleExample 1: Adding velocities on a train
A train moves east at 25 m/s relative to the ground. A passenger rolls a ball at 10 m/s relative to the train. Find the ball's velocity relative to the ground if it is rolled (a) toward the front (east) and (b) toward the back (west).
- Take east as positive. v = +25 m/s (train relative to ground).
- (a) u′ = +10 m/s, so u = u′ + v = 10 + 25 = +35 m/s.
- (b) u′ = −10 m/s, so u = −10 + 25 = +15 m/s. The ball still moves east relative to the ground, but more slowly than the train.
(a) 35 m/s east; (b) 15 m/s east.
Worked exampleExample 2: Transforming an event
Frame S′ moves at v = 25 m/s in the +x direction relative to S, and the origins coincide at t = 0. At t = 2.0 s, a passenger at x′ = 40 m drops a cup. Where does the drop happen in frame S?
- Use x = x′ + vt.
- x = 40 m + (25 m/s)(2.0 s) = 40 m + 50 m = 90 m.
- In Galilean physics t = t′, so the time is 2.0 s in both frames.
The event is at x = 90 m, t = 2.0 s in S (and x′ = 40 m, t′ = 2.0 s in S′).
Worked exampleExample 3: Accelerations agree
A cart accelerates on the train from u′ = 2.0 m/s to u′ = 5.0 m/s in 1.5 s. The train moves at a constant 25 m/s. Find the cart's acceleration in each frame.
- Train frame: a′ = Δu′/Δt = (5.0 − 2.0)/1.5 = 2.0 m/s².
- Ground frame: u goes from 2.0 + 25 = 27 m/s to 5.0 + 25 = 30 m/s. a = (30 − 27)/1.5 = 2.0 m/s².
- The constant v cancels in the subtraction, so a = a′. Since the mass is the same, F = ma has the same form in both frames.
2.0 m/s² in both frames. This is why Newton's laws pass Galileo's test: they are the same in every inertial frame.
The problem with light
In the 1860s, James Clerk Maxwell combined the laws of electricity and magnetism into four equations. They predicted electromagnetic waves that travel at
c = 1/√(μ₀ε₀) ≈ 3.00 × 10⁸ m/s
which matched the measured speed of light. Light was an electromagnetic wave. But this raised a deep question: 3.00 × 10⁸ m/s relative to what?
According to Galilean addition, if a spacecraft moving at 0.5c shines a light forward, a ground observer should measure the light at c + 0.5c = 1.5c. Then Maxwell's equations, which give exactly c, could be true in only one special frame. Mechanics obeyed the principle of relativity, but electromagnetism apparently did not.
Most physicists of the 1800s concluded that light must travel through an invisible medium, the luminiferous ether, and that c is the speed of light relative to the ether. Measuring Earth's motion through the ether should then be possible. The next lesson, The Ether and the Michelson–Morley Experiment, tells what happened when scientists tried.
Key ideas
- A frame of reference is a set of rulers and clocks. Every measurement belongs to a frame.
- Galilean (Newtonian) physics: x = x′ + vt, t = t′, u = u′ + v.
- Accelerations and forces are the same in all inertial frames, so Newton's laws obey the principle of relativity.
- Galilean addition says light's measured speed should depend on the observer's motion. Maxwell's equations say it is c. Something has to give.
Just for funGalileo's ship cabin came with butterflies and a fish bowl. Modern physicists do the same thought experiment in a windowless spaceship, mostly because butterflies are hard to fit in a calculation.
Hands-on activityGalileo's cabin, at home
You need: a balled-up pair of socks (soft, so nothing breaks), a notebook, and either a car driven by an adult or a smooth elevator.
Safety: you are a passenger, buckled in, with the driver concentrating only on driving. Do this only on a quiet, straight road at steady speed, never on a highway, and keep the toss small (about 20 cm). In an elevator, stay away from the doors.
- Inertial frame. While the car moves at a steady speed on a straight road (or the elevator moves steadily between floors), toss the sock ball gently straight up and catch it. Where does it land relative to your hand? Repeat three times.
- Do the same toss when the car is stopped (or the elevator is still). Can you feel or see any difference?
- Non-inertial frame. Ask the driver to tell you before they brake gently or turn. (Do not toss during this; just hold the sock ball loosely on your open palm.) Which way does it roll? Which way does your body lean?
- Write two or three sentences: Could you tell, from the toss alone, whether the car was moving at constant speed or stopped? What changed when the car accelerated?
What to look for: at constant velocity the ball goes straight up and down relative to you, exactly as when parked. That is Galileo's principle. When braking, the ball rolls forward and you lean forward, though nothing pushes you forward: the car is a non-inertial frame and you feel a fictitious force.
Your turn
1Define a frame of reference in your own words.
A coordinate system (rulers) together with synchronized clocks, used by an observer to give every event a position and a time.
2Classify each frame as inertial or non-inertial: (a) a plane cruising at constant velocity, (b) a car going around a curve at constant speed, (c) a rocket firing its engines in deep space, (d) a hockey puck sliding on smooth ice (the puck's own frame).
(a) inertial; (b) non-inertial (its velocity changes direction, so it accelerates); (c) non-inertial (accelerating); (d) inertial (very nearly; friction slows it only slightly).
3A cyclist rides north at 6.0 m/s. A bus passes her going north at 15 m/s relative to the ground. What is the velocity of the bus relative to the cyclist?
Use u′ = u − v with v = cyclist's velocity: 15 − 6.0 = 9.0 m/s north.
4A passenger walks toward the back of a train at 1.5 m/s relative to the train. The train moves forward at 30.0 m/s. Find the passenger's velocity relative to the ground.
u = u′ + v = −1.5 + 30.0 = 28.5 m/s forward.
5Frame S′ moves at 12 m/s in the +x direction relative to S; origins coincide at t = 0. An event happens at x′ = −20 m, t = 5.0 s. Find x.
x = x′ + vt = −20 + (12)(5.0) = 40 m (at t = 5.0 s).
6State Galileo's principle of relativity.
The laws of mechanics are the same in all inertial frames, so no mechanical experiment inside a closed, uniformly moving frame can tell whether it is moving or at rest.
7What value of the speed of light do Maxwell's equations predict, and why was this a problem for Galilean relativity?
c = 1/√(μ₀ε₀) ≈ 3.00 × 10⁸ m/s, with no mention of any frame. Galilean addition says observers moving relative to each other should measure different light speeds, so Maxwell's equations could only hold in one special frame (thought to be the ether frame).
More practice: warm-up
1What is an event? Give an example.
Something that happens at a single place and instant, for example a camera flash going off.
2In Galilean physics, if a clock on a train reads 3.0 s, what does a synchronized clock on the ground read at the same moment?
3.0 s. Newton assumed absolute time: t = t′.
3A boat moves at 4.0 m/s relative to the water in a river flowing at 1.5 m/s. What is its speed relative to the shore going downstream?
4Name the medium that 19th-century physicists believed carried light waves.
The luminiferous ether (often just 'the ether').
5Why do we feel pushed sideways in a car going around a curve, even though no object pushes us outward?
The car is a non-inertial (accelerating) frame. Our body tends to keep moving in a straight line (inertia), and from inside the car this looks like a fictitious outward force.
More practice: on your own
1A jet flies at 250 m/s relative to the ground. A flight attendant pushes a cart toward the front at 1.2 m/s relative to the plane. Find the cart's speed relative to the ground and relative to a second jet flying the same direction at 230 m/s.
Ground: 250 + 1.2 = 251.2 m/s. Second jet: 251.2 − 230 = 21.2 m/s (forward). Velocities relative to different frames differ by the frames' relative velocity.
2A train moving at 20 m/s passes a platform at t = 0 (origins together). A passenger at x′ = 15 m claps at t = 4.0 s. A second clap happens at x′ = 15 m, t = 6.0 s. How far apart are the two claps in each frame?
Train frame: both at x′ = 15 m, so 0 m apart. Ground: x₁ = 15 + 80 = 95 m, x₂ = 15 + 120 = 135 m, so 40 m apart. The distance between events depends on the frame, even in Galilean physics.
3Show that the Galilean transformation gives the same acceleration in both frames.
u = u′ + v with v constant. Then Δu = Δu′ (v cancels) and Δt = Δt′ (absolute time), so a = Δu/Δt = Δu′/Δt′ = a′.
4A spacecraft moves at 1.0 × 10⁸ m/s toward a star. According to Galilean addition, what speed would the crew measure for the star's light coming toward them? What would an observer at rest relative to the star measure?
Galilean: 3.0 × 10⁸ + 1.0 × 10⁸ = 4.0 × 10⁸ m/s for the crew, and 3.0 × 10⁸ m/s for the star's observer. (The next lessons show the crew actually measures 3.0 × 10⁸ m/s too.)
5Explain why a lab on Earth's surface is only approximately inertial.
Earth rotates on its axis and orbits the Sun, so a point on the surface is always accelerating (centripetal acceleration of about 0.03 m/s² at the equator from rotation). These accelerations are small compared with g, so we usually ignore them.
Challenge
1Two cars approach each other on a straight road: car A at 22 m/s east, car B at 18 m/s west (both relative to the ground). (a) What is B's velocity relative to A? (b) A passenger in A tosses a ball east at 5.0 m/s relative to A. What is its velocity relative to B?
Take east +. (a) u′ = u − v = (−18) − (+22) = −40 m/s, i.e. 40 m/s west. (b) Ball relative to ground: 22 + 5.0 = 27 m/s. Relative to B (v = −18): 27 − (−18) = 45 m/s east.
2A sound wave travels at 343 m/s relative to still air. A train moves at 30 m/s and sounds its horn. What speed does a ground observer measure for the sound travelling forward? What about a passenger on the train? Why was light expected to behave the same way?
Sound travels at 343 m/s relative to the air, which is at rest relative to the ground, so the ground observer measures 343 m/s whatever the train does. A passenger moving at 30 m/s into the wave would measure 343 − 30 = 313 m/s for the forward-going sound. Physicists expected light to be a wave in a medium (the ether) too, so its measured speed should depend on the observer's motion through the ether.
3The principle of relativity says no experiment can detect uniform motion. Explain why Maxwell's equations, if combined with the Galilean transformation, seem to break this principle.
Maxwell's equations predict light speed c in all directions. Under Galilean transformations, a moving observer would measure c − v ahead and c + v behind, so Maxwell's equations would be true only in one frame. By measuring light speeds in different directions, you could then detect your own uniform motion relative to that frame, which breaks the principle. Either the principle of relativity fails for light, or Maxwell's equations are wrong, or the Galilean transformation is wrong. Einstein chose the third option.
Quiz
1. Which of these is an inertial frame of reference?
2. A truck moves at 20 m/s east. A ball is thrown west at 8 m/s relative to the truck. What is the ball's velocity relative to the ground (Galilean)?
3. In the Galilean transformation, which quantity is assumed to be the same in every inertial frame?
4. What does Galileo's principle of relativity state?
5. Why did Maxwell's prediction of c ≈ 3.00 × 10⁸ m/s create a problem?
6. A frame S′ moves at 10 m/s in +x relative to S. An event occurs at x′ = 30 m when t = 3.0 s (origins together at t = 0). Where is it in S?
Big idea (F1.1, F2.1): before Einstein, physics assumed absolute time and the simple velocity rule u = u′ + v. This lesson sets up that 'common sense' carefully, so your teen can see exactly which assumption special relativity changes.
How to help: Ask your teen to explain, using a real car or bus ride, why a passenger and a person on the sidewalk disagree about the speed of a coffee cup but agree about how fast it falls when dropped. If you do the activity in a car, the adult driver should focus only on driving; the student tosses only a soft object, and only on a quiet road.
Common difficulties:
- Sign errors in velocity addition. Choose a positive direction first and give every velocity a sign.
- Mixing up u (object relative to S), u′ (object relative to S′) and v (S′ relative to S). Writing a word under each symbol helps.
- Thinking 'the ground is really at rest'. In physics, no inertial frame is special; the ground is just convenient.
- Forgetting that special relativity applies to inertial (non-accelerating) frames only.