Learning Objectives
2 objectivesBy the end of this note, you should be able to:
- 1.4.ADescribe the reference frame of a given observer.
- 1.4.BDescribe the motion of objects as measured by observers in different inertial reference frames.
Choosing a Reference Frame
A reference frame is the coordinate system an observer uses to measure position, velocity, and displacement. Every measurement in physics depends on this choice.
Think of standing on a train platform watching a train pass. You measure the train moving at 30 m/s east. A passenger on that train measures themselves as stationary. Neither observer is wrong — the direction and magnitude of every measured quantity depend on the chosen reference frame.
A reference frame is inertial if it moves at constant velocity (including zero). Newton’s laws hold in all inertial frames. A frame that accelerates is non-inertial, and objects in that frame appear to experience fictitious forces.
MisconceptionStudents often assume one reference frame gives the “true” answer. No inertial frame is preferred; all inertial frames are equally valid for applying Newton’s laws.
Exam TipState your chosen reference frame explicitly before solving any relative-motion problem.
Converting Between Reference Frames
Measurements from one inertial reference frame can be converted to another reference frame using vector addition of velocities.
Key Equations
Relative velocity (Galilean):
$${\vec{v}}_{AC}={\vec{v}}_{AB}+{\vec{v}}_{BC}$$
Variables:
- ${\vec{v}}_{AC}$ = velocity of object A relative to frame C (m/s)
- ${\vec{v}}_{AB}$ = velocity of object A relative to frame B (m/s)
- ${\vec{v}}_{BC}$ = velocity of frame B relative to frame C (m/s)
SI unit: m/s
ProportionalityThe observed velocity of A in frame C is the vector sum of A’s velocity in frame B and B’s velocity in frame C.
Reference sheet status: Derived equation — know how to obtain from vector addition of velocities.
The subscript notation works like a chain: the inner subscripts (B in ${\vec{v}}_{AB}$ and ${\vec{v}}_{BC}$) must match. The outer subscripts give the result (${\vec{v}}_{AC}$). This pattern extends to any number of frames.
One-dimensional case. Choose a positive direction first. Velocities in that direction are positive; velocities opposite are negative. Combining motions then reduces to simple addition or subtraction of signed values.
Two-dimensional case. Resolve each velocity into perpendicular components. Add corresponding components separately, then find the resultant magnitude and direction.
$${v}_{AC}=\sqrt{{v}_{AC,x}^{2}+{v}_{AC,y}^{2}}$$
$$\theta ={tan}^{-1}\left(\frac{{v}_{AC,y}}{{v}_{AC,x}}\right)$$
A crucial result: the acceleration of any object is the same as measured from all inertial reference frames. Because inertial frames differ only by a constant velocity, the time derivative of that constant is zero. So ${\vec{a}}_{A,\text{frame B}}={\vec{a}}_{A,\text{frame C}}$ for any two inertial frames B and C. Forces, masses, and Newton’s second law therefore give identical results in every inertial frame.
Examiner InsightFRQs often present a moving platform (boat, train, conveyor belt) and ask for velocity relative to the ground. Write the subscript chain first, then add vectors.
Exam TipAlways define your positive direction and state both magnitude and direction in relative-velocity answers.
Worked Example: River Crossing
A boat heads due north across a river at 4.0 m/s relative to the water. The river flows due east at 3.0 m/s relative to the ground. Determine the boat’s velocity relative to the ground.
Define positive directions: north = +y, east = +x.
Equation used
$${\vec{v}}_{BG}={\vec{v}}_{BW}+{\vec{v}}_{WG}$$
Given
$${v}_{BW,y}=4.0\text{ m/s}$$
$${v}_{BW,x}=0\text{ m/s}$$
$${v}_{WG,x}=3.0\text{ m/s}$$
$${v}_{WG,y}=0\text{ m/s}$$
Working — components of ${\vec{v}}_{BG}$
$${v}_{BG,x}=0+3.0=3.0\text{ m/s}$$
$${v}_{BG,y}=4.0+0=4.0\text{ m/s}$$
Magnitude:
$${v}_{BG}=\sqrt{(3.0{)}^{2}+(4.0{)}^{2}}=\sqrt{9.0+16.0}=\sqrt{25.0}=5.0\text{ m/s}$$
Direction:
$$\theta ={tan}^{-1}\left(\frac{4.0}{3.0}\right)=53.1^{\circ}$$
$${v}_{BG}=5.0\text{ m/s at }53.1^{\circ}\text{ north of east}$$
The river’s eastward current pushes the boat off its northward heading. An observer on the ground sees the boat travel at 5.0 m/s along a path angled 53° north of east.
| Feature | Same across inertial frames? | Changes between frames? |
|---|---|---|
| Acceleration | Yes | No |
| Velocity | No | Yes — shifts by the relative velocity of the frames |
| Displacement (over a time interval) | No | Yes — differs by ${\vec{v}}_{\text{rel}}\times \Delta t$ |
| Force (Newton’s second law) | Yes | No |
QUICK RECAP
Key Points
- A reference frame is the coordinate system an observer uses for measurement.
- An inertial reference frame moves at constant velocity (or is at rest).
- Newton’s laws hold in all inertial reference frames.
- Velocity measurements depend on the observer’s reference frame.
- Relative velocity: ${\vec{v}}_{AC}={\vec{v}}_{AB}+{\vec{v}}_{BC}$ (subscript chain rule).
- Inner subscripts must match when chaining velocity vectors.
- In 1D, choose a positive direction and use signed values.
- In 2D, resolve velocities into components and add separately.
- Acceleration is the same in all inertial reference frames.
- Forces and masses are identical across inertial frames.
- Always state your chosen reference frame before solving.
- Always define the positive direction in relative-motion problems.
- State both magnitude and direction for every velocity answer.
CAN I…? PROGRESS CHECK
Self-Assessment
- Can I identify and describe the reference frame of a given observer?
- Can I convert a velocity measurement from one inertial frame to another using vector addition?
- Can I solve a two-dimensional relative velocity problem using component resolution?
- Can I explain why acceleration is invariant across inertial reference frames?
- Can I justify that Newton’s second law gives identical results in any inertial frame?
- Can I apply the subscript chain method to set up a relative velocity equation?