Relative Speed
Relative speed tracks the rate at which two moving objects close the distance between them or pull apart, depending on their travel directions.
Fundamental Principles
Opposite Vector Direction
When two objects move toward each other or away from each other in opposite directions. Their relative speed is the sum of their individual speeds ($S_1 + S_2$).
Same Vector Direction
When two objects move in the exact same direction on parallel paths. Their relative speed is the difference between their individual speeds ($S_1 - S_2$).
Essential Formulation Tips
- Always double-check the directional path indicators (+ or -) before choosing your relative speed formula.
- In pursuit scenarios (like a police car chasing a thief), the initial gap between the objects serves as your total target distance.
Shortcut Execution Techniques
- The Overtake Framework: The total time required for a faster object to catch up to a slower moving object is calculated as: $\text{Time to Overtake} = \frac{\text{Initial Distance Gap}}{\text{Relative Speed Difference}}$.
Contextual Inquiries (FAQs)
Q: Why do we add individual speeds together when two vehicles travel toward each other?
A: Because both vehicles are actively working to close the distance gap at the same time, increasing their combined rate of approach.
Example Breakdown: Resolving Same-Direction Pursuit Targets
Classic same-direction pursuit problem configuration.Identify the initial distance gap: $\text{Distance} = 200 \text{ meters}$.
Determine relative motion direction: Both are moving in the same direction, so subtract their speeds.
Calculate relative speed difference: $\text{Relative Speed} = 11 \text{ km/h} - 10 \text{ km/h} = 1 \text{ km/h}$.
Convert relative speed to meters per second: $1 \times \frac{5}{18} = \frac{5}{18} \text{ m/s}$.
Set up the time pursuit equation: $\text{Time} = \frac{\text{Initial Distance Gap}}{\text{Relative Speed}}$.
Substitute your values and multiply by the reciprocal fraction: $\text{Time} = 200 \div \frac{5}{18} = 200 \times \frac{18}{5}$.
Perform final reduction: $40 \times 18 = 720 \text{ seconds}$. Convert to minutes: $\frac{720}{60} = 12 \text{ minutes}$.
Conclusion: The security guard catches the thief in 12 minutes.
Relative Motion Calculations
Practice finding crossover points and pursuit times under different directional paths.
Q1. Two towns are 150 km apart. Car A starts from one town traveling at 20 km/h, and Car B starts from the other at 30 km/h, moving toward each other. How many hours until they meet?