Comprehensive Study Notes: Time, Speed, and Distance (TSD) Concepts and Applications

Time, Speed, and Distance (TSD)

Basic Relations

Understanding the relationship between time, speed, and distance is fundamental in solving motion problems. The core equations are:

• Distance (D) is the product of Speed (S) and Time (T):

$D = S \times T$

• Speed (S) is the ratio of distance to time:

$S = \frac{D}{T}$

• Time (T) is the ratio of distance to speed:

$T = \frac{D}{S}$

Unit Conversion

• 1 km/hr = $\frac{5}{18}$ m/sec

• 1 m/sec = $\frac{18}{5}$ km/hr

Cases: Constant Variables

• Distance Constant: $S_1 T_1 = S_2 T_2$

• Time Constant: $\frac{D_1}{S_1} = \frac{D_2}{S_2}$

• Speed Constant: $\frac{D_1}{T_1} = \frac{D_2}{T_2}$

Meeting Problems

When two objects move towards each other or away, their relative speeds and meeting times are crucial:

• Speed Ratio: $\sqrt{t_2 : t_1}$

• Meeting Time: $\sqrt{t_1 t_2}$

Relative Speed

• Same Direction: $|S_1 - S_2|$

• Opposite Direction: $S_1 + S_2$

• Catching Time: $\text{Catching Time} = \frac{\text{Distance}}{\text{Relative Speed}}$

Trains

Scenario	Formula
Train passes pole/man	$\text{Time} = \frac{\text{Length}}{\text{Speed}}$
Train passes platform/bridge	$\text{Time} = \frac{\text{Train Length} + \text{Platform Length}}{\text{Speed}}$
Two trains, opposite direction	$\text{Time} = \frac{L_1 + L_2}{S_1 + S_2}$
Two trains, same direction	$\text{Time} = \frac{L_1 + L_2}{|S_1 - S_2|}$
Train passes man in moving train	$\text{Time} = \frac{\text{Length}}{\text{Relative Speed}}$

Special Train Results

• Same length trains, same direction: $\text{Time} = \frac{2 t_1 t_2}{t_1 - t_2}$

• Same length trains, opposite direction: $\text{Time} = \frac{2 t_1 t_2}{t_1 + t_2}$

• Length of train: $\text{Length} = P \times t_1 = T \times t_2$

Average Speed

• General: $\text{Avg speed} = \frac{\text{Total distance}}{\text{Total time}}$

• Equal distances at speeds x, y: $\text{Avg speed} = \frac{2xy}{x + y}$

• Equal parts with speeds $s_1, s_2, ..., s_n$: $\text{Avg speed} = \frac{n}{\frac{1}{s_1} + \frac{1}{s_2} + ... + \frac{1}{s_n}}$

Speed Change Cases

• Distance: $\text{Distance} = \frac{xy}{|x - y|} \times \text{(change in time)}$

• If speed becomes A/B times: $\text{Original time} = \frac{A}{|A - B|} \times \text{given time}$

Boat & Stream

• Speed of boat = x, Speed of stream = y

• Downstream (D): $x + y$

• Upstream (U): $x - y$

• Speed of stream (if D and U known): $y = \frac{D - U}{2}$

• Avg speed (go & return): $\frac{2xy}{x^2 - y^2}$

• If equal distances upstream & downstream: $\text{Time} = \frac{2d}{x^2 - y^2}$

Boat Special Results

• Same time upstream $d_1$ & downstream $d_2$: Boat speed : Stream speed = $(d_1 + d_2) : (d_1 - d_2)$

Race

• If A beats B by d meters: Speed ratio = $x : (x - d)$

• If race length = L and times $t_A, t_B$: $d = B$'s speed $\times (t_B - t_A)$

• Start problems: $L = (c - b) L / c$

Circular Race

• Time to meet again: $\text{Time} = \frac{\text{Circumference}}{\text{Relative speed}}$

Golden Rule

In TSD problems, always convert into relative speed and unit time first.