Reference Frames and Relative Motion
| English | Chinese | Pinyin |
|---|---|---|
| reference frame | 参考系 | cān kǎo xì |
| relative velocity | 相对速度 | xiāng duì sù dù |
How fast are you really walking?
- Stroll down the aisle of a moving train at a gentle $2\ \text{km/h}$.
- To the passenger beside you, that is your speed.
- To someone standing outside, you are doing $102\ \text{km/h}$.
- Both are right -- speed depends on who is watching.
Reference frames
- A reference frame 参考系 is the point of view -- the observer -- you measure motion against.
- Velocity is always measured relative to some frame.
- Change the frame and the numbers change, even for the same motion.
The observer or point of view against which you measure motion is called a ____ frame.
A reference frame is the viewpoint; all velocities are stated relative to one.
Relative velocity
- Relative velocity 相对速度 is found by adding velocities as vectors.
- Your velocity relative to the ground = your velocity in the train + the train's velocity.
- Same direction adds; opposite subtracts; at an angle, use components.
You walk forward at $1.5\ \tfrac{\text{m}}{\text{s}}$ in a train moving at $25\ \tfrac{\text{m}}{\text{s}}$. Your speed relative to the ground (in m/s)?
Same direction, so add them, $1.5 + 25 = 26.5\ \tfrac{\text{m}}{\text{s}}$.
Now you walk backward at $1.5\ \tfrac{\text{m}}{\text{s}}$ in the same $25\ \tfrac{\text{m}}{\text{s}}$ train. Your speed relative to the ground (in m/s)?
Opposite directions subtract, $25 - 1.5 = 23.5\ \tfrac{\text{m}}{\text{s}}$.
The addition rule
- The velocity of A relative to C is $\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$.
- Chain the frames: A-relative-to-B, then B-relative-to-C.
- Reverse a subscript pair and you flip the sign: $\vec{v}_{BA} = -\vec{v}_{AB}$.
Relative velocity
Add a boat's velocity across a river to the current's velocity to get its velocity relative to the bank.
To combine velocities across frames, you...
$\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$ -- a vector sum (which becomes plain adding or subtracting along one line).
Pick a smart frame
- Some problems are far easier in the right frame.
- A collision looks simplest in the frame of the center of mass.
- River-crossing problems are easiest split into "relative to water" and "water relative to ground."
There is one single "true" velocity for an object, the same for every observer.
Velocity is always relative to a frame -- different observers measure different velocities for the same object.
Select all true statements about relative motion.
Velocity is frame-dependent and frames add as vectors; no velocity is absolute.
You walk forward at $2\ \tfrac{\text{m}}{\text{s}}$ inside a train moving at $30\ \tfrac{\text{m}}{\text{s}}$.
- Relative to the ground you move at $2 + 30 = 32\ \tfrac{\text{m}}{\text{s}}$.
- Walk backward instead and it is $30 - 2 = 28\ \tfrac{\text{m}}{\text{s}}$.
There is no single "true" velocity -- only velocity relative to a chosen frame. A ball dropped in a moving bus falls straight down to you but follows a curve to someone outside. Always state (or assume) the frame before quoting a velocity.
A reference frame is the observer you measure against; velocity is always relative to one. Combine frames with vector addition: $\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$ (relative velocity). No velocity is absolute -- choosing a smart frame can make a hard problem easy.