AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/1/1-4)
Unit 1 · Topic 1.4
1.4 Reference Frames and Relative Motion
Every measurement of motion is made from some reference frame, and observers in different frames can disagree about position and velocity. In one dimension you switch frames by adding or subtracting velocities, and all inertial observers agree on acceleration.
Key terms
- reference frame
- observer
- relative velocity
- inertial reference frame
What a reference frame is
A reference frame is the point of view you measure from: an origin and a set of axes that move along with an observer. A passenger sitting on a train is at rest in the train's frame but moving at 30 m/s in the ground's frame. Both descriptions are correct.
Change the frame and the numbers change: the same car can be moving forward at 25 m/s, sitting still or drifting backward, depending on who measures it. Unless a problem says otherwise, assume measurements are made from the ground.
Combining velocities in one dimension
The velocity an observer sees is the object's velocity in some other frame combined with the velocity of that frame. With subscripts, read v_AB as "velocity of A relative to B." Then:
v_AC = v_AB + v_BC
For a ball thrown on a train: v_ball,ground = v_ball,train + v_train,ground. Signs carry the directions. A ball thrown backward at 8 m/s from a train moving forward at 20 m/s moves forward at 12 m/s relative to the ground.
Reversing the subscripts flips the sign: v_BA = −v_AB. Taking forward as positive, if you're gaining on a car ahead at 5 m/s, that car is moving at −5 m/s relative to you.
Switching frames can make problems easier. To find how long a fast car takes to catch a slower one 100 m ahead, work in the slow car's frame: the gap closes at the difference of their velocities, so with a 5 m/s difference it takes 100 ÷ 5 = 20 s.
Inertial frames and acceleration
An inertial reference frame is one that isn't accelerating: it's at rest or moving at constant velocity. In an inertial frame, Newton's first law holds, so an object with no net force keeps its velocity (2.4).
All inertial observers measure the same acceleration for an object. If a ball is dropped in a train moving at constant velocity, a passenger and someone on the platform disagree about its velocity, but both measure 9.8 m/s² downward.
In an accelerating frame, like a car that brakes hard, objects seem to lurch forward with nothing pushing them. That's a sign the frame is noninertial. AP problems use inertial frames unless they say otherwise.
Same motion, different pictures
Changing frames changes the shape of a motion description. A passenger who drops a ball sees it fall straight down. Someone on the platform sees it keep the train's forward velocity while it falls, so it traces a curved path. On a position–time graph, switching to a frame moving at constant velocity changes the slope of the graph by the frame's velocity, but not its curvature, since acceleration doesn't change.
What's not on the exam
AP Physics 1 limits relative velocity calculations to one dimension. You won't be tested on two-dimensional problems such as a boat heading at an angle across a flowing river or an airplane flying in a crosswind.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Moving walkway
An airport walkway is 60 m long and moves at 0.75 m/s. A traveler walks at 1.5 m/s relative to the walkway. How long does the trip take if she walks (a) with the walkway's motion and (b) against it?
Show the solutionHide the solution
- Step 1: Take the walkway's direction as positive. v_walkway,ground = +0.75 m/s.
- Step 2: (a) v_traveler,ground = v_traveler,walkway + v_walkway,ground = 1.5 + 0.75 = 2.25 m/s. Time = 60 m ÷ 2.25 m/s ≈ 26.7 s.
- Step 3: (b) Now v_traveler,walkway = −1.5 m/s, so v_traveler,ground = −1.5 + 0.75 = −0.75 m/s. She still moves against the walkway, at 0.75 m/s relative to the ground. Time = 60 m ÷ 0.75 m/s = 80 s.
Answer: (a) about 27 s; (b) 80 s
- Example 2Calculator allowed
Relative velocity of two cars (classic trap)
Car A drives east at 25 m/s. Find the velocity of car B relative to car A if car B drives (a) east at 18 m/s and (b) west at 18 m/s. Take east as positive.
Show the solutionHide the solution
- Step 1: Use v_BA = v_BG + v_GA, where G is the ground. Since v_AG = +25 m/s, v_GA = −25 m/s.
- Step 2: (a) v_BA = (+18) + (−25) = −7 m/s. To a passenger in A, car B drifts backward (west) at 7 m/s.
- Step 3: (b) v_BA = (−18) + (−25) = −43 m/s. To a passenger in A, car B rushes past westward at 43 m/s.
- Step 4: The trap is adding the speeds in (a) or subtracting them in (b). Let the signs do the work.
Answer: (a) 7 m/s west (−7 m/s); (b) 43 m/s west (−43 m/s)
- Example 3Calculator allowed
A coin dropped in a moving elevator
An elevator rises at a constant 2.0 m/s. A passenger lets go of a coin from rest in her hand. Use g = 9.8 m/s² and take up as positive. Find the coin's velocity 0.30 s later (a) relative to the elevator and (b) relative to the building, and compare the accelerations the two observers measure.
Show the solutionHide the solution
- Step 1: (a) In the elevator's frame the coin starts at rest: v = 0 + (−9.8)(0.30) ≈ −2.9 m/s.
- Step 2: (b) In the building's frame the coin starts with the elevator's velocity, +2.0 m/s: v = 2.0 + (−9.8)(0.30) ≈ −0.94 m/s.
- Step 3: Check with the frame rule: v_coin,building = v_coin,elevator + v_elevator,building = −2.94 + 2.0 = −0.94 m/s. ✓
- Step 4: Both observers measure the same acceleration, 9.8 m/s² downward, because the elevator's frame moves at constant velocity and is inertial.
Answer: (a) about 2.9 m/s downward; (b) about 0.94 m/s downward; both see 9.8 m/s² downward
Common mistakes
- Adding speeds without signs. Write every velocity with a sign in one coordinate system before combining.
- Mixing up v_AB and v_BA. They're equal in size and opposite in sign.
- Thinking a dropped object on a moving train lands behind the dropping point. In an inertial train it lands directly below, because it keeps the train's horizontal velocity.
- Believing observers in different inertial frames measure different accelerations. Velocity depends on the frame; acceleration doesn't.
On the exam
- Expect conceptual questions comparing what two observers see, often with sketches of paths or graphs. Say which frame each description is from.
- A common setup asks for one object's velocity relative to another, or the time for one to catch the other. Converting to the frame of one object often makes it a simple constant-velocity problem.
Connected topics
Videos
Check yourself
4 questions on 1.4 Reference Frames and Relative Motion. Pick an answer to see if you got it, and why.
A train moves east at 15 m/s relative to the ground. A passenger walks toward the back of the train at 2.0 m/s relative to the train. What is the passenger's velocity relative to the ground?
Car A moves east at 25 m/s and car B moves west at 20 m/s on a straight road. What is the velocity of car B as measured by the driver of car A?
A passenger on a train moving at constant velocity drops a ball. A person standing on the platform watches it fall. Air resistance is negligible. Which statement is correct?
A boat moves at 5.0 m/s relative to the water, heading directly upstream in a river that flows at 2.0 m/s. How long does it take the boat to travel 300 m upstream relative to the shore?
0 of 4 answered