🚗 Time, Speed and Distance

Day 2 | ⏱️ Part of 2-Hour Commercial Math Block | TCS Interview Prep

📌 Subtopics

• Basic speed, distance, time relationship
• Average speed
• Relative speed (same/opposite direction)
• Trains (crossing poles, platforms, other trains)
• Boats and streams (upstream/downstream)
• Races and circular tracks

📖 Comprehensive Notes

1. Basic Speed, Distance, Time Relationship

The fundamental formula relating distance ($D$), speed ($S$), and time ($T$) is: $Distance = Speed \times Time$

• Unit Conversions:
  • To convert $\text{km/h}$ to $\text{m/s}$, multiply by $\frac{5}{18}$.
  • To convert $\text{m/s}$ to $\text{km/h}$, multiply by $\frac{18}{5}$.
• Proportionalities:
  • If Distance is constant, $Speed \propto \frac{1}{Time}$ ($\frac{S_1}{S_2} = \frac{T_2}{T_1}$).
  • If Time is constant, $Distance \propto Speed$ ($\frac{D_1}{D_2} = \frac{S_1}{S_2}$).
  • If Speed is constant, $Distance \propto Time$ ($\frac{D_1}{D_2} = \frac{T_1}{T_2}$).

Example: A car runs at $72\text{ km/h}$. How many meters does it travel in $15$ seconds? Speed in $\text{m/s} = 72 \times \frac{5}{18} = 20\text{ m/s}$. Distance $= Speed \times Time = 20 \times 15 = 300\text{ meters}$.

2. Average Speed

Average speed is not the simple average of speeds.

• General Formula: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$
• Special Case (Equal Distances): If two equal distances are traveled at speeds $u$ and $v$: $\text{Average Speed} = \frac{2uv}{u+v}$

Example: A man travels the first half of a journey at $30\text{ km/h}$ and the remaining half at $20\text{ km/h}$. Find his average speed. Since distances are equal: $\text{Average Speed} = \frac{2 \times 30 \times 20}{30 + 20} = \frac{1200}{50} = 24\text{ km/h}$.

3. Relative Speed

Relative speed is the speed of one object with respect to another moving object.

• Opposite Directions: If two objects move at speeds $S_1$ and $S_2$ towards or away from each other: $\text{Relative Speed} = S_1 + S_2$
• Same Direction: If two objects move in the same direction: $\text{Relative Speed} = |S_1 - S_2|$

Example: Two trains start at the same time from stations A and B, $300\text{ km}$ apart, towards each other at $60\text{ km/h}$ and $40\text{ km/h}$. When will they meet? Relative Speed $= 60 + 40 = 100\text{ km/h}$. Time to meet $= \frac{\text{Distance}}{\text{Relative Speed}} = \frac{300}{100} = 3$ hours.

4. Problems on Trains

Trains have a significant length that must be considered.

• Crossing a point object (e.g., pole, tree, standing man): $\text{Distance} = \text{Length of train } (L)$
• Crossing a long object (e.g., platform, bridge, tunnel of length $P$): $\text{Distance} = L + P$
• Crossing another train (lengths $L_1$ and $L_2$): $\text{Distance} = L_1 + L_2$
  • Note: Divide distance by the appropriate Relative Speed.

Example: A train $150\text{ m}$ long is running at $54\text{ km/h}$. How long will it take to cross a platform of length $250\text{ m}$? Total Distance $= 150 + 250 = 400\text{ m}$. Speed $= 54 \times \frac{5}{18} = 15\text{ m/s}$. Time $= \frac{400}{15} = 26.67$ seconds.

5. Boats and Streams

• Let the speed of the boat in still water $= u\text{ km/h}$.
• Let the speed of the stream (current) $= v\text{ km/h}$.
• Downstream Speed ($D_s$): Moving with the flow of the stream. $D_s = u + v$
• Upstream Speed ($U_s$): Moving against the flow of the stream. $U_s = u - v$
• Derived Formulas:
  • Speed of boat in still water: $u = \frac{D_s + U_s}{2}$
  • Speed of stream: $v = \frac{D_s - U_s}{2}$

Example: A boat goes $36\text{ km}$ downstream in $2$ hours and $24\text{ km}$ upstream in $3$ hours. Find the stream speed. $D_s = \frac{36}{2} = 18\text{ km/h}$. $U_s = \frac{24}{3} = 8\text{ km/h}$. Stream speed $v = \frac{18 - 8}{2} = 5\text{ km/h}$.

6. Races and Circular Tracks

• Races: “A gives B a start of $x$ meters” means B starts $x$ meters ahead of A. “A beats B by $y$ seconds” means A finishes $y$ seconds before B.
• Circular Tracks: If two runners run on a circular track of length $L$ with speeds $a$ and $b$:
  • Time to meet at the starting point for the first time: $\text{LCM}\left(\frac{L}{a}, \frac{L}{b}\right)$.
  • Time to meet anywhere on the track for the first time: $\frac{L}{a+b}$ (opposite direction) or $\frac{L}{|a-b|}$ (same direction).

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

Distance = Speed × Time
Speed    = Distance / Time
Time     = Distance / Speed

Average Speed (equal distances) = 2ab / (a+b)
Relative Speed (same dir)  = |S1 - S2|
Relative Speed (opp dir)   = S1 + S2

Trains: Time = (Length1 + Length2) / Relative Speed
Boats: Downstream = (Boat + Stream), Upstream = (Boat - Stream)

Unit Conversion: km/hr × 5/18 = m/s

🔗 Related Topics

• Time and Work — Rate-based problem solving with similar structure
• Averages — Average speed is a common concept shared with averages
• Equations — Setting up distance/speed equations
• Directions and Distance — Distance calculation in reasoning context
• Clocks and Calendars — Relative speed concept applies to clock hands