Solving Relative Motion Problems with Elevators — Step-by-StepRelative motion problems involving elevators are a classic application of kinematics and reference frames. These problems often appear in introductory physics courses and standardized tests because they neatly illustrate how observed velocities and accelerations depend on the observer’s frame of reference. This article presents a clear, step-by-step approach to solving such problems, illustrated with examples and common pitfalls.
1. Understand the scenario and choose reference frames
Begin by carefully reading the problem. Ask:
- Who are the observers? (e.g., a person inside the elevator, a stationary observer on the ground, or another moving object)
- What quantities are given? (velocities, accelerations, directions, times, distances)
- What is being asked? (relative speed, acceleration, displacement, time)
Common reference frames:
- Ground (inertial frame): typically the building floor or an external observer.
- Elevator (non-inertial if accelerating): the frame attached to the elevator.
- A moving object within or outside the elevator (e.g., a ball thrown inside, or a person walking in the elevator).
Always define a positive direction (upwards or downwards) and stick with it.
2. Identify known quantities and sign convention
List known velocities and accelerations with proper signs. For example:
- Elevator moving upward at 2 m/s → velocity ve = +2 m/s (if up is positive).
- Person walks upward inside elevator at 1 m/s relative to elevator → v_person|e = +1 m/s.
- If the elevator accelerates downward at 0.5 m/s^2 → a_e = −0.5 m/s^2.
Be consistent: if up is positive, downward values are negative.
3. Use relative velocity relations
For velocities, the basic relation is: v_A|B = v_A|ground − v_B|ground, where v_A|B denotes velocity of A relative to B.
Two commonly used rearrangements:
- v_A|ground = v_A|B + v_B|ground
- v_A|B = v_A|ground − v_B|ground
Examples:
- Person walking inside an upward-moving elevator: v_person|ground = v_person|e + v_e|ground.
- Ball thrown upward inside the elevator (measured relative to the elevator) can be converted to ground-frame initial velocity using the same additivity.
4. Account for accelerations when frames accelerate
If the elevator accelerates, velocities measured in the elevator are related to ground-frame velocities by the same additive relation for instantaneous velocities. For accelerations, a_A|ground = a_A|B + a_B|ground, where a_A|B is acceleration of A relative to B.
However, when working inside an accelerating (non-inertial) frame, fictitious (pseudo) forces appear in Newton’s second law if you choose to analyze dynamics from the elevator frame. For kinematics problems that only ask for velocities or displacements, it’s usually easier to work in the inertial ground frame.
5. Convert kinematic equations between frames
Once you have initial velocities in the chosen (usually ground) frame, use standard kinematic equations:
- v = v0 + a t
- x = x0 + v0 t + ⁄2 a t^2
- v^2 = v0^2 + 2 a (x − x0)
If an object is observed inside the elevator with a given relative velocity and the elevator accelerates, convert the initial relative velocity to ground frame using v_ground = v_relative + v_elevator (at that instant), then apply kinematics in ground frame.
6. Example 1 — Simple relative speed (no acceleration)
Problem: An elevator moves upward at 3 m/s. A person inside walks upward at 1.5 m/s relative to the elevator. What is the person’s speed relative to the ground?
Solution:
- Choose up as positive.
- v_e = +3.0 m/s; v_person|e = +1.5 m/s.
- v_person|ground = v_person|e + v_e = 1.5 + 3.0 = 4.5 m/s.
7. Example 2 — Ball thrown inside accelerating elevator
Problem: An elevator accelerates upward at 0.8 m/s^2. Inside, a passenger throws a ball upward at 4.0 m/s relative to the elevator. Find the ball’s initial acceleration and initial velocity relative to the ground, and the maximum height reached above the elevator floor (ignore air resistance).
Solution:
- At the instant of release, elevator acceleration a_e = +0.8 m/s^2 (up positive).
- Relative initial velocity v_ball|e = +4.0 m/s.
- Convert to ground frame: v_ball|ground = v_ball|e + v_e|ground. If the elevator has instantaneous velocity v_e = Ve (not given), you can express ground velocity in terms of Ve; but acceleration conversion is independent: a_ball|ground = a_ball|e + a_e. If the ball moves freely after release, its acceleration relative to ground is gravity g downward: a_ball|ground = −g. The relation a_ball|e = a_ball|ground − a_e indicates the ball’s acceleration as seen from elevator frame equals −g − a_e = −(g + a_e). That affects how the passenger perceives the ball’s motion.
- Maximum height relative to elevator floor: use elevator-frame kinematics with effective acceleration a_eff = −(g + a_e). Then h_max = v0^2 / [2 (g + a_e)] = 4.0^2 / [2 (9.8 + 0.8)] ≈ 16 / 19.2 ≈ 0.833 m.
Note: If you prefer ground-frame calculation you must know instantaneous elevator velocity to convert v0; the elevator-frame method avoids that by using effective acceleration.
8. Example 3 — Two people moving in different frames
Problem: Elevator moves down at 2 m/s. Inside, a person A walks up at 1.2 m/s relative to the elevator. A second person B is standing on the roof of the elevator and walks forward (relative to ground) at 1.0 m/s upward. What is B’s speed relative to A?
Solution:
- Let up be positive. v_e = −2.0 m/s. v_A|e = +1.2 m/s → v_A|ground = 1.2 + (−2.0) = −0.8 m/s (A moves downward relative to ground).
- v_B|ground = +1.0 m/s.
- v_B|A = v_B|ground − v_A|ground = 1.0 − (−0.8) = 1.8 m/s upward relative to A.
9. Common pitfalls and tips
- Sign mistakes: Always choose and stick to a positive direction.
- Confusing relative-to-ground and relative-to-elevator velocities — label every velocity with its reference frame.
- Using non-inertial frames for dynamics without including fictitious forces leads to errors; for kinematics questions you can often remain in the ground frame.
- If an elevator’s instantaneous velocity is not given but its acceleration is, use the elevator frame for displacement/maximum-height problems where that cancels out via effective acceleration.
- For problems involving rotation (e.g., elevator cables, pulleys) include rotational constraints separately.
10. Quick problem-solving checklist
- Define frames and positive direction.
- List known quantities with frame labels.
- Convert relative velocities/accelerations to the chosen frame using v_A|B = v_A|ground − v_B|ground and a_A|B = a_A|ground − a_B|ground.
- Solve kinematics/dynamics in inertial frame if possible; if using accelerating frame, include pseudo-forces for dynamics.
- Check signs and units; assess limiting cases for sanity.
Relative motion in elevators is a small, tidy playground for practicing frame choice and transformation. With consistent sign conventions and careful labeling of each velocity/acceleration by frame, these problems become straightforward algebra and substitution.
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