Time, Speed & Distance Guide & Practice

Core Foundations

Fundamental Formulas

• Basic Relation: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
• Distance: $\text{Distance} = \text{Speed} \times \text{Time}$
• Time: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Unit Conversions

• km/hr to m/s: Multiply by $\frac{5}{18}$
• m/s to km/hr: Multiply by $\frac{18}{5}$
• Other Conversions:
  • 1 mile = 1.609 km = 1760 yards = 5280 feet
  • 1 yard = 3 feet
  • 1 km = 1000 m
  • 1 hr = 60 min = 3600 sec

Proportionality Rules (When one variable is constant)

1. Time is constant: Distance is directly proportional to Speed ($D \propto S$).
2. Speed is constant: Distance is directly proportional to Time ($D \propto T$).
3. Distance is constant: Speed is inversely proportional to Time ($S \propto \frac{1}{T}$).
   • If speeds are in the ratio $a:b$, time taken will be in the ratio $b:a$.

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Thematic Deep-Dive

1. Average Speed

• General Formula: $\text{Average Speed} = \frac{\text{Total Distance Travelled}}{\text{Total Time Taken}}$
• Constant Distance: If an object covers equal distances at speeds $x$ and $y$:
  • $\text{Average Speed} = \frac{2xy}{x+y}$
• Constant Time: If an object travels for equal time intervals at speeds $x$ and $y$:
  • $\text{Average Speed} = \frac{x+y}{2}$

2. Relative Speed

• Opposite Direction: Speeds are added ($S_1 + S_2$).
• Same Direction: Speeds are subtracted ($|S_1 - S_2|$).
• Meeting Point: If two objects start at the same time towards each other, the time taken to meet is $\frac{\text{Initial Distance}}{\text{Relative Speed}}$.

3. Problems on Trains

• Crossing a Point Object (Pole, Man): $\text{Distance} = \text{Length of Train}$.
• Crossing a Long Object (Platform, Bridge): $\text{Distance} = \text{Length of Train} + \text{Length of Object}$.
• Two Trains Crossing: $\text{Distance} = \text{Length of Train 1} + \text{Length of Train 2}$.
  • Use relative speed based on direction.

4. Boats and Streams

• Still Water Speed ($u$): Speed of boat in stationary water.
• Stream Speed ($v$): Speed of the current.
• Downstream Speed ($D$): $u + v$ (with the flow).
• Upstream Speed ($U$): $u - v$ (against the flow).
• Derived Formulas:
  • $u = \frac{D + U}{2}$
  • $v = \frac{D - U}{2}$

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