Inertial Frames and Non-inertial Frames
Key idea: Learn what makes a frame inertial, how Newton’s laws change in accelerating/rotating frames, and how to use (and interpret) fictitious forces.
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The core idea
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Learning objectives
- state that a frame of reference is a set of coordinates that can be used to determine positions and times of events in that frame
- show an understanding that Newton’s laws of motion are obeyed in all inertial frames of reference
An inertial frame is a frame of reference where Newton’s laws hold in their standard form (no extra “made-up” forces needed). This lesson sets the language you’ll need for Galilean transformations and the centre of mass frame later in the H3 syllabus.
1. Definitions (Must Know)
- Frame of reference: A set of coordinates + a clock used to assign positions and times to events.
- Event: Something that happens at a particular place and time (e.g. “the ball hits the floor”), described by (x, y, z, t) in a chosen frame.
- Inertial frame: A frame in which a free particle (net force = 0) moves with constant velocity (Newton’s first law holds), hence Newton’s laws apply without modification.
- Non-inertial frame (enrichment): A frame that is accelerating and/or rotating relative to an inertial frame; Newton’s laws can still be used, but only if fictitious forces are introduced.
- Symbols used in this lesson: u relative speed (m s⁻¹), h height (m), g ≈ 9.81 m s⁻² near Earth, m mass (kg), vector a₀ frame acceleration (m s⁻²), v speed (m s⁻¹), r radius (m), a_c centripetal acceleration (m s⁻²).
2. Key Ideas (What Earns Marks)
- Newton’s laws are obeyed in all inertial frames. There is no “preferred” inertial frame for mechanics.
- Different inertial observers can disagree on position, velocity, and the shape of a path, but they should still be able to explain the motion using the same physical laws.
- A practical “test” of inertial-ness: in your frame, if an object with no net force does not move in a straight line at constant speed, your frame is not inertial (or it is only an approximation).
Quick comparison (use a table when asked to compare):
| Feature | Inertial frame | Non-inertial frame |
|---|---|---|
| Free particle (net force = 0) | Straight line, constant speed | Appears to accelerate/curve |
| Newton’s laws | Standard form | Need fictitious forces to keep F = ma usable |
| Typical examples | Train at constant velocity (approx.) | Accelerating bus, rotating carousel |
3. Detailed Explanations
A. What a frame of reference really is
A frame of reference is not just “where you stand”. You must specify:
- an origin and axes (coordinates), and
- how time is measured (a clock in that frame).
Only after that can you talk meaningfully about “the position at time t”.
B. Why constant-velocity frames are inertial (Galilean idea)
Suppose frame S' moves at constant velocity u relative to inertial frame S along the x-axis. The Galilean relations are:
Differentiate with respect to time:
So acceleration is the same in both frames. If Newton’s second law holds in S: ∑ F = ma then it also holds in S' because the same real forces act on the object and a' = a.
C. The “ball tossed in a plane/train” example (what changes, what doesn’t)
- In the plane (inertial, approximately): The passenger throws the ball straight up. They see a vertical path because the ball has no horizontal velocity relative to them.
- On the ground (also inertial, approximately): The ball has the plane’s horizontal velocity as well as vertical motion under gravity, so the path is a parabola.
Both observers still explain the motion using the same forces (mainly weight, ignoring air resistance), and both use the same constant vertical acceleration g.
4. Common Mistakes
- Writing “inertia frame” instead of inertial frame.
- Thinking an inertial frame must be at rest; any frame moving at constant velocity relative to an inertial frame is also inertial.
- Concluding “parabolic path” means “different laws”; the path shape depends on the observer’s velocity, not on whether Newton’s laws hold.
- Treating the Earth’s surface as perfectly inertial in every situation (it’s often a good approximation, but not exact).
5. Exam Tips
- Use mark-scheme-safe phrasing: “An inertial frame is one in which Newton’s laws of motion are obeyed.”
- If asked to justify “no preferred inertial frame”, describe a closed experiment done entirely inside a constant-velocity vehicle (train/plane) that cannot detect its uniform motion.
- State your assumption when needed: “Air resistance negligible, so the only force is weight, giving constant acceleration g.”
6. Worked Examples
Modelled example 1
Dropping a ball inside a moving train
Problem
Study the worked solution
Classify the frames
Method
The constant-velocity train and ground are treated as inertial frames.Reason
Their relative velocity is constant, so no fictitious force is needed.Working
a_frame = 0Describe the train view
Method
The ball falls straight down with acceleration g.Reason
It has no horizontal velocity relative to the train when released.Working
x' = constant, y = h-(1/2)gt²Describe the ground view
Method
The ball follows a parabola.Reason
It retains horizontal speed u while accelerating vertically.Working
x = ut, y = h-(1/2)gt²
Guided practice 2
Acceleration seen in two inertial frames
Problem
Try this before viewing the solution
Hints
Hint 1: start from velocity
Hint 2: differentiate
View solution step by step
Transform velocity
Method
v' = v-u.Reason
The frames differ by a constant Galilean boost.Working
v' = v-uDifferentiate
Method
a' = a.Reason
du/dt = 0 for constant relative velocity.Working
a' = dv'/dt = dv/dt-du/dt = a
Common misconception 3
A frictionless puck in an accelerating bus (fictitious force)
Learner claim
Try this before viewing the solution
View solution step by step
Start in the ground frame
Method
aₓ = 0.Reason
No horizontal real force acts on the frictionless puck.Working
∑ Fₓ = 0 ⇒ aₓ = 0Transform acceleration
Method
a'ₓ = -2.0 m s⁻².Reason
The bus frame accelerates at + a₀ relative to the inertial frame.Working
a'ₓ = aₓ-a₀ = 0-2.0 = -2.0 m s⁻²Introduce the frame force
Method
F_(fict,x) = -1.0 N.Reason
A fictitious force -ma₀ restores the usual F = ma' form within the bus frame.Working
F_(fict,x) = -(0.50)(2.0) = -1.0 N
Examiner practice 4
“Effective gravity” in an accelerating elevator
Examination question
Try this before viewing the solution
View solution step by step
Classify the frame
1 markMethod
The elevator frame is non-inertial.Reason
It accelerates relative to an inertial frame.Working
a₀ ≠ 0Set the effective direction
1 markMethod
The fictitious acceleration is downward, adding to gravity.Reason
It acts opposite the elevator’s upward acceleration.Working
g_eff = g + a₀Calculate
1 markMethod
g_eff = 11.31 m s⁻².Reason
Add the two downward acceleration magnitudes.Working
g_eff = 9.81 + 1.5 = 11.31 m s⁻²
Self-mark with the mark scheme
Compare your response with each mark point. Select a point only when your response contains that evidence.
Self-mark frame classification, direction/relation and value.
Challenge 5
A turning car (rotating frame intuition)
Independent transfer
Try this before viewing the solution
Hints
Hint 1: separate inertial and rotating views
View solution step by step
Calculate centripetal acceleration
Method
a_c = 4.5 m s⁻² inward.Reason
Circular motion requires a_c = v²/r toward the centre.Working
a_c = 15²/50 = 4.5 m s⁻²Find the frame-force magnitude
Method
F_fict = 270 N.Reason
The rotating-frame term has magnitude ma_c.Working
F_fict = (60)(4.5) = 270 NGive its direction
Method
The fictitious force points outward.Reason
It is opposite the inward acceleration of the rotating car frame.Working
vector F_fict points away from the centre
7. Mind Stretchers
Mind stretcher 1: Is a lab on Earth an inertial frame?Extension
Earth rotates, so a lab frame is (strictly) non-inertial. When is it still a good approximation to treat it as inertial?
One way to think about it
Compare the rotation-related accelerations to g. If the “extra” accelerations (e.g. centripetal due to Earth’s rotation) are tiny compared to g or other accelerations in the problem, treating the lab frame as inertial is a good approximation.
Mind stretcher 2: A quick “inertial test” you can do inside a sealed vehicleExtension
You are inside a sealed train carriage with no windows. Propose a simple experiment you could do inside the carriage that can detect whether the carriage is accelerating, but cannot detect whether it is moving at constant velocity.
One good answer
Release a small object (or roll a ball) and observe its motion relative to the carriage. If the carriage is accelerating, a free object will appear to accelerate opposite the carriage’s acceleration (straight-line constant-velocity motion fails), revealing a non-inertial frame. If the carriage moves at constant velocity, Newton’s first law still holds and the same internal experiment cannot distinguish it from rest.
8. Optional/Enrichment: Non-inertial Frames and Fictitious Forces
If your frame has acceleration vector a₀ relative to an inertial frame, objects can appear to “accelerate for no reason” in your frame. One way to keep Newton’s second law usable is to introduce a fictitious force: vector F_fict = -m vector a₀
This is useful intuition, but the detailed treatment of fictitious forces is better placed at university mechanics level:
Next step
Return to the Frames of Reference hub, or continue in sequence to Galilean Transformations.
Continue with the next resource in this course.
Course and syllabus information
- Course
- GCE A-Level H3 Physics
- Edition
- GCE A-Level H3 Physics 2027