Multi-stage & relative motion
A new stage changes the rule of motion, not the identity of the particle. The end state of one stage must be handed to the next stage with the same position scale and a clearly managed clock.
Carry the end state into the next stage
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- Finish the current stage. Find the boundary time, velocity and position.
- Write the handover. State which values become the next stage's initial data.
- Choose the clock. Keep global time , or define a local time such as . Never mix the two silently.
Key idea
Velocity is normally continuous when a driving rule changes: its value immediately after the boundary equals its value immediately before. It can jump only when the model includes an instantaneous event, meaning an idealised event that takes no time, such as an impact.
Write down each stage before choosing equations
Worked example
Acceleration followed by constant speed
A particle starts from rest and accelerates at for 4 s, then continues at constant velocity for 6 s.
Answer
Common mistake
Do not use 10 s inside the first-stage equation. The first stage lasts 4 s; the second lasts 6 s; the whole journey lasts 10 s.
Compare positions on one shared coordinate system
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- Meeting or catching: solve .
- Use the same origin, positive direction and global time for both particles.
- While B remains ahead of A, their separation is , so . An interior stationary value of the gap therefore occurs when . Also check the interval endpoints and any meeting time, then compare the gap values to identify the true minimum.
- At an idealised collision, position is continuous but velocity may change instantly. A velocity–time graph may therefore have a jump.
Your turnIndependent check [4 marks]
Particle A starts at position 0 and moves right at . At the same instant, particle B starts at position 40 m and moves left at . Find when and where they meet.
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Answer