Frames of reference

Compare inertial and non-inertial frames, apply Galilean transformations and use the centre-of-mass frame to simplify collisions.

GCE A-Level H3 Physics 2027
4 lessons

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Before you begin

A frame of reference is the set of axes and the clock an observer uses to give every event a position and a time. The lessons start by telling inertial frames from accelerating ones, then use the Galilean transformation to move between frames, and finish by solving one-dimensional collisions in the centre-of-mass (COM) frame.

Be comfortable with: Newton’s laws and momentum conservation in collisions, using signed velocities.

Learning goals

State that a frame of reference is a set of coordinates for locating the positions and times of events
Explain that Newton's laws of motion hold in every inertial frame of reference
Recall and use the Galilean transformation to relate observations in different inertial frames
Explain that the centre-of-mass (zero-momentum) frame is the inertial frame in which a system's total momentum is zero
Solve one-dimensional collisions using velocities relative to the centre of mass

Syllabus statements covered

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
recall and apply the Galilean transformation equations to solve problems relating observations in different inertial frames of reference
show an understanding that the centre of mass frame (or zero momentum frame) is the inertial frame in which the total linear momentum of the system is zero
solve one-dimensional collision problems by considering velocities relative to the centre of mass of the system (i.e. in the zero-momentum frame)

Lessons

Work through them in order.

Practise and check

Or choose

Topic reference

Quick reference

Idea	Relation	Notes
Galilean transformation	x' = x-ut, t' = t	S' moves at + u along x relative to S
velocity	v'ₓ = vₓ-u	speeds much less than c
acceleration	a'ₓ = aₓ	the same in all inertial frames
fictitious force in a frame accelerating at vector af	vector F_fict = -m vector af	not a physical interaction
COM velocity	vector V_cm = (∑ mᵢ vector vᵢ)/(∑ mᵢ)	total momentum is zero in the COM frame
elastic collision, COM frame	v'₁ = -u'₁, v'₂ = -u'₂	each velocity reverses
objects that stick, COM frame	v' = 0	lab velocity = V_cm

Core knowledge to remember

In an inertial frame, a particle with no resultant force stays at rest or moves at constant velocity, and Newton’s laws hold in their usual form.
Any frame moving at constant velocity, with non-rotating axes, relative to an inertial frame is also inertial. No inertial frame is preferred.
An accelerating or rotating frame is non-inertial. Using F = ma there needs fictitious forces, which come from the frame’s acceleration.
Inertial observers disagree about positions, velocities and the shapes of paths, but agree about accelerations and forces.
The COM frame of an isolated system is inertial and has zero total momentum, before and after a collision.
To solve a collision in the COM frame: transform the velocities, apply the collision rule, then transform back to the lab.

Common mistakes

Writing x' = x + ut because the direction in which S' moves was never stated.
Using speeds instead of signed velocities to find V_cm.
Treating an accelerating frame as inertial.
Confusing the COM frame with one object’s rest frame.
Giving COM-frame velocities when the question asks for lab-frame velocities.
Comparing kinetic energies measured in different frames.