Time, Speed & Distance
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Speed, Distance, Time & Unit Conversion
Basic relation
S = \frac{D}{T}, \quad D = S \times T
km/h to m/s
\text{km/h} \times \frac{5}{18} = \text{m/s}
Equal-distance average
\bar{S} = \frac{2ab}{a + b}
General average speed
\bar{S} = \frac{D_1 + D_2}{T_1 + T_2}
Relative Speed
Relative speed
S_{\text{rel}} = S_1 \pm S_2
Meeting time
t = \frac{\text{initial gap}}{S_1 + S_2}
Overtake time
t = \frac{\text{gap or combined length}}{S_1 - S_2}
Time after meeting (opposite starts)
\frac{\text{speed of other}}{\text{own speed}} = \frac{\text{time to B from meet}}{\text{time to A from meet}}
Trains Crossing Poles, Platforms & Trains
Pole / man
L_{\text{train}} = S \times t
Platform / bridge
L_{\text{train}} + L_{\text{platform}} = S \times t
Train vs train
L_1 + L_2 = S_{\text{rel}} \times t
Two-equation extraction
\frac{t_{\text{platform}}}{t_{\text{pole}}} = \frac{L + P}{L}
Boats & Streams
Effective speeds
u = b + s, \quad v = b - s
Boat & stream from trips
b = \frac{u + v}{2}, \quad s = \frac{u - v}{2}
Time for two legs
t = \frac{d_1}{b + s} + \frac{d_2}{b - s}
Swimmer/boat carried by stream
\text{drift} = s \times \text{time}
Races & Handicaps
Beating margin in metres
\frac{S_A}{S_B} = \frac{L}{L - x}
Beating margin in time
\text{B's remaining time} = t \Rightarrow S_B = \frac{x}{t}
Speeds from 'by x m or t s'
S_B = \frac{x}{t}, \quad S_A = S_B \cdot \frac{L}{L - x}
Start handicap
\text{B runs } L - \text{start}