AP Physics 1- 1.4 Reference Frames and Relative Motion- Study Notes- New Syllabus
AP Physics 1-1.4 Reference Frames and Relative Motion – Study Notes
AP Physics 1-1.4 Reference Frames and Relative Motion – Study Notes -AP Physics 1 – per latest Syllabus.
Key Concepts:
- Motion in Different Inertial Reference Frames
- Relative Motion in One and Two Dimensions
Motion in Different Inertial Reference Frames
Observers in different inertial reference frames may measure different positions and velocities of the same object, but they will always agree on its acceleration (if only real forces act).
- This is because acceleration depends on the net force acting on the object, and Newton’s laws hold true in all inertial frames.
Galilean Relativity (valid at low speeds compared to the speed of light):
- If an object has velocity \( v \) in one inertial frame, and the second frame moves with velocity \( v_{\text{frame}} \) relative to the first, then:
- \( v’ = v – v_{\text{frame}} \)
Key point: Motion is always relative to the chosen frame of reference.
Example :
A passenger in a train moving at \( 15 \, \text{m/s} \) throws a ball forward at \( 10 \, \text{m/s} \) relative to the train. Find the velocity of the ball relative to the ground.
▶️Answer/Explanation
Step (1): Use Galilean transformation → \( v’ = v + v_{\text{train}} \).
Step (2): \( v’ = 10 + 15 = 25 \, \text{m/s} \).
Final Answer: Observer on the ground sees the ball moving at 25 m/s.
Example :
A car accelerates at \( 2 \, \text{m/s}^2 \) relative to the ground. An observer sitting inside another car moving at constant velocity beside it measures the acceleration of the first car. What value is obtained?
▶️Answer/Explanation
Step (1): Both observers are in inertial frames.
Step (2): Acceleration is the same in all inertial frames if only real forces act.
Final Answer: Acceleration measured = 2 m/s² in both frames.
Relative Motion in One and Two Dimensions
Relative motion describes how the motion of an object appears to one observer compared to another moving observer.
- Key principle: Motion is always relative — there is no universal state of rest.
Relative Motion in One Dimension (1D)
If observer A sees an object moving with velocity \( v_{OA} \), and observer B is moving with velocity \( v_{BA} \) relative to A, then:
- \( v_{OB} = v_{OA} – v_{BA} \)
- Directions are handled by assigning positive/negative signs.
Example
A train moves east at \( 20 \, \text{m/s} \). A passenger walks west inside the train at \( 2 \, \text{m/s} \) relative to the train. Find the velocity of the passenger relative to the ground.
▶️Answer/Explanation
Step (1): Take east as positive. Train velocity = \( +20 \, \text{m/s} \). Walking velocity = \( -2 \, \text{m/s} \).
Step (2): Relative to ground → \( v_{PG} = v_{PT} + v_{TG} = -2 + 20 = 18 \, \text{m/s} \).
Final Answer: Passenger moves east at 18 m/s relative to ground.
Relative Motion in Two Dimensions (2D)
Velocities in 2D are treated as vectors.
- Rule: \( \vec{v}_{OB} = \vec{v}_{OA} – \vec{v}_{BA} \).
- Vector addition (using components or geometry) is required.
Common situations:
- Boat crossing a river (current + boat velocity).
- Airplane flying with wind.
Example
A boat moves with velocity \( 5 \, \text{m/s} \) due north in still water. The river flows east at \( 3 \, \text{m/s} \). Find the boat’s velocity relative to the ground.
▶️Answer/Explanation
Step (1): Represent velocities as vectors:
- \( \vec{v}_{BN} = (0, 5) \, \text{m/s} \) (north).
- \( \vec{v}_{RE} = (3, 0) \, \text{m/s} \) (east).
Step (2): Net velocity relative to ground → \( \vec{v} = (3, 5) \, \text{m/s} \).
Step (3): Magnitude → \( v = \sqrt{3^2 + 5^2} = \sqrt{34} \approx 5.83 \, \text{m/s} \).
Step (4): Direction (angle east of north) → \( \theta = \tan^{-1} \left(\dfrac{3}{5}\right) \approx 31^\circ \).
Final Answer: Boat’s velocity relative to ground = 5.83 m/s at 31° east of north.