Unit 4: Time, Distance, Trains, Boats and Streams

I. Orientation

Time–distance problems are governed by the relationship between the distance travelled, the time taken, and the speed of motion. If motion is uniform, the same distance is covered in equal intervals of time; if speed changes, average speed must be used to describe the complete journey.

• Fundamental relationship: Distance equals speed multiplied by time.
• Measurement convention: Distance, speed, and time must be expressed in compatible units before calculation.
• Uniform motion: Speed remains constant throughout the journey.
• Relative motion: The observed speed depends on whether two objects move in the same or opposite directions.
• Train convention: A train crossing a point covers its own length; crossing a platform or bridge requires covering the combined lengths.
• Water-current convention: A boat’s speed in still water combines with or opposes the stream speed depending on its direction.
• Direction convention: Downstream means moving with the current; upstream means moving against the current.

II. Concept of Time, Speed, and Distance — The basic motion relationship

Time, speed, and distance are three interdependent quantities used to measure motion. Knowing any two determines the third, provided the units are consistent.

A. Concept of time, speed, and distance

The point of this concept is to express motion through one standard equation and its rearrangements.

• Distance: Distance is the total path covered by an object. It is usually measured in metres (m) or kilometres (km).
• Time: Time is the duration of motion, measured in seconds (s), minutes (min), or hours (h).
• Speed: Speed is the distance covered per unit time.

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

• Symbols: Let S represent distance, v represent speed, and t represent time.
• Uniform-motion equation:

S = vt

• Concrete interpretation: A car moving at 60 km/h covers 60 km in 1 hour and 120 km in 2 hours.
• Dimension check: If v = km/h and t = h, then S = km; the time unit cancels correctly.
• Zero condition: If distance is non-zero and time is zero, the calculated speed is undefined; in ordinary motion, a positive distance requires positive time.
• Graphical meaning: In a distance–time graph, the slope represents speed. A steeper line indicates greater speed.

B. Conversion of units and proportionality

The point of unit conversion is to put all quantities into a common measurement system before applying the motion formula.

• Basic conversions:
  • 1 kilometre = 1000 metres
  • 1 hour = 60 minutes = 3600 seconds
  • 1 minute = 60 seconds
• Speed conversion:

1 km/h = 1000 m / 3600 s = 5/18 m/s
1 m/s = 18/5 km/h

• Example conversion: 72 km/h = 72 × 5/18 = 20 m/s.
• Proportionality with fixed distance: For a fixed distance, time is inversely proportional to speed.

t ∝ 1/v

Thus, if speed changes from v₁ to v₂ while distance remains constant:

t₁v₁ = t₂v₂

• Numerical effect: If speed increases by 25%, the new speed is 1.25v; the time becomes t/1.25 = 0.8t, so time decreases by 20%.
• Proportionality with fixed time: For a fixed time, distance is directly proportional to speed.

S ∝ v

A vehicle travelling twice as fast for the same duration covers twice the distance.

C. Applications and limitations

The point of applying the basic formula is to model ordinary journeys while identifying conditions that make the formula insufficient by itself.

• Application: For 180 km travelled in 3 h, speed is 180/3 = 60 km/h.
• Changing speed: If a journey contains different speeds, one cannot generally average the listed speeds directly; total distance must be divided by total time.
• Unit limitation: 60 km/h + 10 m/s is invalid until both speeds are converted to the same unit.
• Distance versus displacement: These problems usually use total distance, not directed displacement; direction becomes important only in relative motion and stream problems.

III. Average Speed — Speed over a complete journey

Average speed represents the uniform speed that would produce the same total distance in the same total time. It is based on totals, not on the ordinary arithmetic mean of speeds.

A. Average speed concept

The point of the average speed concept is to handle journeys made at different speeds or over different time intervals.

• Formal definition: Average speed equals total distance divided by total time.

Average speed = Total distance / Total time

• Symbols: Let Sₜ be total distance, tₜ be total time, and v_avg be average speed.

v_avg = Sₜ / tₜ

• Unequal distances: If distances d₁ and d₂ are covered at speeds v₁ and v₂, then:

v_avg = (d₁ + d₂) / (d₁/v₁ + d₂/v₂)

• Equal distances: If the same distance is covered at speeds v₁ and v₂, then:

v_avg = 2v₁v₂ / (v₁ + v₂)

The harmonic-mean form appears because the time spent on each equal distance is different.

• Equal time intervals: If speeds v₁ and v₂ continue for equal times, average speed is:

v_avg = (v₁ + v₂) / 2

• Worked example: A cyclist travels 30 km at 15 km/h and returns 30 km at 10 km/h. Total distance is 60 km; total time is 2 + 3 = 5 h. Therefore, average speed is 60/5 = 12 km/h, not (15 + 10)/2 = 12.5 km/h.
• Bounds: For positive speeds, average speed lies between the smallest and largest actual speeds.

B. Applications and limitations

The point of this application is to select the correct average-speed formula from the structure of the journey.

• Journey segments: Add all segment distances and all segment times before dividing.
• Stops: If the problem asks for average speed for the entire elapsed journey, stopping time is included in total time. If it asks for average running speed, stationary periods are excluded.
• Round trips: Equal outward and return distances require the equal-distance formula, even when the two speeds differ.
• Unit consistency: A result such as 12 km/h must not be combined with a time in seconds without conversion.
• Physical interpretation: Average speed does not indicate that the object actually moved at that exact speed at every instant; it describes the overall rate.

IV. Relative Speed and Trains — Motion observed between objects

Relative speed measures how quickly the distance between two moving objects changes. It is the central tool for train-crossing problems and for many overtaking or meeting situations.

A. Relative speed concept and application to trains

The point of relative speed is to replace two motions with one effective motion viewed from the other object.

• Opposite directions: When two objects move toward or past each other in opposite directions, their relative speed is the sum.

v_rel = v₁ + v₂

• Same direction: When two objects move in the same direction, relative speed is the difference.

v_rel = |v₁ - v₂|

• Symbols: v₁ and v₂ are the actual speeds; v_rel is their relative speed.
• Meeting time: If two objects are initially D apart:

Time to meet = D / v_rel

Use the sum for opposite directions and the difference for the same direction.

• Train crossing a stationary point: A train of length L crossing a pole, signal, or person covers only L.

Time = L / v

Here L is train length and v is train speed.

• Train crossing a platform or bridge: The train must cover its own length plus the platform or bridge length P.

Time = (L + P) / v

• Two trains in opposite directions: If their lengths are L₁ and L₂, the total distance to clear each other is L₁ + L₂.

Time = (L₁ + L₂) / (v₁ + v₂)

• Two trains in the same direction: If the faster train overtakes the slower train:

Time = (L₁ + L₂) / |v₁ - v₂|

• Worked example: A 150 m train moving at 54 km/h crosses a 300 m platform. Since 54 km/h = 15 m/s, the distance is 450 m, and the time is 450/15 = 30 s.

B. Applications and limitations

The point of train applications is to identify exactly what distance must be covered before the objects are completely separated.

• Complete crossing: “Crosses a pole” means the rear of the train reaches the pole; “crosses a platform” means the rear clears the far end.
• Direction check: Opposite-direction trains use speed addition; trains moving in the same direction use speed subtraction.
• Moving observer: If a person walks inside or beside a train, use the relative speed between the person and the train.
• Length recovery: If a train crosses a pole in t seconds at v m/s, its length is L = vt.
• Unit warning: A train speed stated in km/h must be converted to m/s when lengths are given in metres and time is required in seconds.
• Limitation: These equations assume constant speeds, straight-line motion, and negligible acceleration during the crossing.

V. Boats and Streams — Motion in still water and flowing water

Boat-and-stream problems distinguish the boat’s speed relative to still water from its speed relative to the bank. The current changes the effective speed according to direction.

A. Downstream

The point of downstream motion is to calculate travel with the stream, where the current assists the boat.

• Definition: Downstream travel is movement in the same direction as the water current.
• Speed relationship: If b is the boat’s speed in still water and s is the stream speed, downstream speed is:

v_down = b + s

• Time calculation: For downstream distance D:

t_down = D / (b + s)

Here D is distance and t_down is downstream time.

• Interpretation: A boat with still-water speed 12 km/h in a stream of 3 km/h moves downstream at 15 km/h relative to the bank.
• Worked example: For a 45 km downstream journey at 15 km/h, time is 45/15 = 3 h.
• Stream assistance: The current contributes s to the bank-measured speed; it does not increase the boat’s speed relative to the surrounding water.

B. Upstream

The point of upstream motion is to calculate travel against the stream, where the current reduces the boat’s effective speed.

• Definition: Upstream travel is movement opposite to the direction of the water current.
• Speed relationship: If b is still-water boat speed and s is stream speed, upstream speed is:

v_up = b - s

• Time calculation: For upstream distance D:

t_up = D / (b - s)

• Required condition: The boat must have b > s; otherwise, it cannot make forward progress upstream.
• Recovering component speeds: If downstream speed is D_s and upstream speed is U_s:

b = (D_s + U_s) / 2
s = (D_s - U_s) / 2

• Worked example: If downstream speed is 18 km/h and upstream speed is 10 km/h, still-water speed is (18 + 10)/2 = 14 km/h, while stream speed is (18 - 10)/2 = 4 km/h.
• Time comparison: For the same distance, upstream travel takes longer because b - s < b + s.
• Limitations: The standard model assumes a steady current, constant boat speed in still water, and no effects from wind, turning, loading, or changing river width.