Reference Frames and Relative Motion
1. A reference frame is a decision, not a place
A reference frame is three choices made together: where the origin sits, which direction counts as positive, and what the object is at rest with respect to. Make those three choices and every position, velocity and acceleration on the previous three pages becomes a definite number. Change any one of them and the numbers change.
The usual frame is the ground, and it is so usual that most problems never mention it. That is exactly why it is worth naming out loud. When a problem says a car travels at 25 m/s, it means relative to the road, and the moment a second moving observer enters the problem that unstated assumption is the thing that trips people.
2. The subscript rule, and why it works
Write vAB to mean the velocity of A as measured by B. Read the subscripts in that order and keep them in that order, and relative motion in one dimension stops requiring any cleverness at all.
vAC = vAB + vBC
The inner subscripts match and cancel, and the outer two survive. A person walking on a train has a velocity relative to the train, the train has a velocity relative to the ground, and adding them gives the person relative to the ground.
vPG = vPT + vTG
The second rule is shorter. Swapping the subscripts flips the sign, because if you are moving away from me at four meters per second then I am moving away from you at four meters per second in the other direction.
vAB = − vBA
One motion, two observers
The same scene is drawn twice. On top, a frame fixed to the ground, where the posts stand still. Below, a frame fixed to the train, where the train stands still and the posts stream past. Set the two velocities and watch what each observer measures. Nothing about the physical situation changes when you switch panels. Only the bookkeeping does.
Push the person’s velocity to −1.5 m/s and leave the train at 12 m/s. The person is walking toward the back of the train and is still moving forward at 10.5 m/s as far as the ground is concerned. Walking backward and moving backward are not the same claim, and the frame is what separates them.
3. What changes, and the one thing that does not
Compare two frames moving at a constant velocity with respect to each other. Positions differ. Velocities differ, by exactly the relative velocity of the frames. But look at what happens to the acceleration.
vPA = vPB + vBAand vBA is constant
Take the rate of change of both sides. A constant has no rate of change, so the last term contributes nothing and the two accelerations are equal.
aPA = aPB
Notice what that sentence quietly rules out. In a braking bus your coffee slides forward, and in the bus frame nothing pushed it. The frame is accelerating, so the guarantee does not apply there. Choose the road as your frame and the coffee is doing the least surprising thing in physics: continuing at the velocity it already had while the bus slows underneath it.
Who measures what
A scene, and a question about one observer’s measurement. Three answers are offered. Pick one, and the feedback names the mistake rather than just grading the box.
4. The ball dropped on a moving train
A passenger on a train moving at a steady 30 m/s holds a ball at shoulder height and lets go. Two people describe what happens, and both are right.
The passenger says the ball fell straight down and landed at her feet. In her frame the ball started at rest, and the only thing acting on it was gravity, straight down.
The person on the platform says the ball traveled a curved path, a parabola, moving forward the whole way. In his frame the ball started with a forward velocity of 30 m/s, kept it, and fell at the same time.
They disagree about the shape of the path, the length of the path, and the speed. They agree on two things: the acceleration was 9.8 m/s² downward, and the ball landed at the passenger’s feet. The first agreement is the theorem from section 3. The second is the one that matters, because where the ball lands is a fact about the world, and no choice of frame is allowed to change it.
Check yourself
1. A passenger walks toward the front of a train at 1.4 m/s relative to the train. The train moves at 26 m/s relative to the platform. What is the passenger’s velocity relative to the platform, and relative to the seat she just left?
Relative to the platform, vPG = 1.4 + 26 = 27.4 m/s. Relative to the seat, which is part of the train, 1.4 m/s. The second answer is the one people talk themselves out of. The seat is at rest in the train frame, so the passenger’s velocity relative to it is just her velocity relative to the train.
2. Two cars travel the same way down a road, one at 31 m/s and one at 24 m/s. What is the faster car’s velocity as measured by the slower one? Now turn the slower car around so it comes the other way at the same speed, and answer again.
Same direction: vAB = vAG + vGB = 31 + (−24) = +7 m/s. The driver of the slower car sees the other pull away at walking-pace-plus. Head on, the second car is now at −24 m/s in the road frame, so 31 + 24 = +55 m/s. Nothing on either car changed. Only the frame did, and that is why closing speed is what wrecks people rather than road speed.
3. A swimmer heads straight across a river at 1.2 m/s relative to the water, and the water flows downstream at 0.8 m/s relative to the bank. Explain why she cannot land straight across, using subscripts.
vSB = vSW + vWB. She controls only the first term. The second is added to it whether she likes it or not, so her velocity relative to the bank has a downstream part she never chose. Aiming straight across means being carried downstream. Landing straight across means aiming upstream, so that the two contributions cancel sideways.
4. A bus brakes hard and a bag slides forward along the floor. A student says a forward force pushed the bag. Say what is wrong with that, and name the frame that makes it wrong.
In the road frame nothing pushed the bag forward at all. The bag kept the velocity it already had while the bus slowed underneath it, so the bag moved forward relative to the bus without accelerating relative to the road. The student was reasoning in the bus frame, which is accelerating and therefore not inertial, so the rule that an acceleration needs a force does not hold there. Switch to the road and the problem disappears.
5. Two observers moving at a constant velocity with respect to each other measure the same object. Name one quantity they must disagree about and one they must agree about, and say why in each case.
They disagree about velocity, by exactly vBA, the relative velocity of their frames. They agree about acceleration, because that relative velocity is constant and a constant has no rate of change. Position also differs, and so does the shape of the path. Any event, such as a collision happening or a ball landing on a particular tile, is also agreed on, because an event is not a measurement of motion but a fact about what occurred.
Unit 2, Force and Translational Dynamics. Section 3 of this page quietly did the work: every inertial observer agrees on acceleration. Unit 2 is what you get when you ask what causes it.