Questions: Relative Motion and Moving Reference Frames

Question 1

A drone moves at 3 m/s east relative to a boat. The boat moves at 5 m/s east relative to the ground. What is the drone's velocity relative to the ground?

• A. 2 m/s east — subtract the frame velocity from the particle velocity
• B. 3 m/s east — the drone's velocity relative to the boat is its true velocity
• C. 8 m/s east — add the relative velocities as vectors
• D. 5 m/s east — only the frame's velocity matters to a ground observer

Answer: (C)

v_drone/ground = v_drone/boat + v_boat/ground = 3 + 5 = 8 m/s east. This is the fundamental velocity addition law for non-rotating frames. Each velocity is a vector quantity, so direction matters — here they are both eastward, so we add magnitudes directly.

Question 2

A ball rolls radially outward at constant speed along a rotating turntable. An observer on the turntable sees the ball curve sideways. An observer on the ground sees it travel in a straight line. A student concludes: 'There must be a real sideways force on the ball since it curves in the turntable frame.' What is the correct analysis?

• A. The student is correct — a real sideways force acts on the ball in all frames
• B. The sideways curvature is the Coriolis acceleration — a fictitious effect that appears in the rotating frame but corresponds to no real force
• C. Both observers are wrong — objects always curve in rotating systems
• D. The ground-frame observer would also see the ball curve sideways due to Earth's rotation

Answer: (B)

In a rotating reference frame, a particle moving relative to the frame experiences a Coriolis acceleration (2ω × v_rel) that appears as a sideways deflection — but there is no corresponding real force. In the ground frame (inertial), the ball travels in a straight line with no sideways force at all. The Coriolis 'force' is a fictitious force that arises because the rotating frame is non-inertial. This is the same effect responsible for cyclone rotation on Earth.

Question 3

The velocity of a particle relative to a fixed frame equals its velocity relative to a moving (translating, non-rotating) frame plus the velocity of the moving frame itself.

• T. True
• F. False

Answer: True

True. This is the fundamental velocity addition law: v_P/A = v_P/B + v_B/A. It holds as a vector equation — both magnitudes and directions must be accounted for. This decomposition — particle velocity in the moving frame plus the frame's own velocity — is the foundation of all relative motion analysis for translating frames.

Question 4

The Coriolis term 2ω × v_rel in the rotating-frame acceleration equation accounts for the rotation of the reference frame itself.

• T. True
• F. False

Answer: False

False. The Coriolis term specifically accounts for the additional acceleration arising because a particle moving inside the rotating frame continuously enters regions with different velocities due to the frame's spin — it is the interaction between the particle's velocity relative to the frame and the frame's angular velocity. The frame's own rotation contributes separately through the centripetal term ω × (ω × r) and, if angular velocity changes, an α × r term. Each term in the full expression has a distinct physical origin.

Question 5

Why is it essential to specify which reference frame a velocity derivative is taken in, and what error arises if you mix frames?

Model answer: The time derivative of a position vector differs depending on which frame's basis vectors are used. In a rotating frame, the basis vectors themselves rotate, contributing additional terms when differentiated. If you take a velocity measured in frame B and add or subtract a velocity whose derivative was taken in frame A, the result is a meaningless mixture. The error manifests as missing Coriolis, centripetal, and angular-acceleration terms in the acceleration expression — leading to incorrect equations of motion.

This is the most common trap in rotating-frame problems. The shorthand 'velocity of P' is ambiguous unless you specify: velocity of P as measured by whom? A derivative taken in frame A and one taken in frame B are related by d/dt|_A = d/dt|_B + ω × (·), where ω is the angular velocity of B relative to A. Skipping this step means your acceleration equation will be wrong by exactly the missing correction terms.