Time, Speed & Distance

Time, Speed & Distance

Concept

The big secret of this chapter? There’s only one real formula: Speed = Distance ÷ Time. Everything else is just this formula applied cleverly.

Average Speed — The Trap

Here’s where students lose marks: average speed is NOT (v₁ + v₂) ÷ 2. That only works when traveling for the SAME TIME at each speed. When you travel the same DISTANCE at two different speeds, the correct formula is:

$$v_{avg} = \frac{2 v_1 v_2}{v_1 + v_2}$$

Why? Because you spend more time at the slower speed (covering the same distance), so the average skews toward the slower speed.

Trains — The Length Factor

A train isn’t a point — it has length! When a train crosses:

• A pole: The distance covered = train’s own length. Time = Length ÷ Speed.
• A platform: Distance = train length + platform length.
• Another train: Distance = sum of both train lengths. Use relative speed if they’re moving.

Boats in Water — The Current Effect

A boat has a speed in still water (call it u). The stream flows at speed v.

• Downstream (with current): effective speed = u + v
• Upstream (against current): effective speed = u – v

You can find both:

• u = (downstream + upstream) ÷ 2
• v = (downstream – upstream) ÷ 2

Relative Speed — Same or Opposite Direction?

When two objects move:

• Same direction: Relative speed = difference of speeds (the faster “gains” on the slower)
• Opposite direction: Relative speed = sum of speeds (they “close the gap” faster)

Key Formulas

Formula	Use
Speed = Distance/Time	Core relationship
Average Speed (same dist) = 2v₁v₂/(v₁+v₂)	When covering equal distances at v₁ and v₂
Average Speed (same time) = (v₁+v₂)/2	When traveling equal times
Downstream speed = u + v	Boat with stream
Upstream speed = u – v	Boat against stream
Relative speed (same dir) =	v₁ – v₂
Relative speed (opp dir) = v₁ + v₂	Trains/objects going opposite ways
Time to cross train = (L₁+L₂)/relative speed	When two trains cross

Worked Example

Q: Two trains 150 m and 200 m long run at 40 km/hr and 30 km/hr in the same direction. How long to completely pass each other?

Step 1: Relative speed = 40 – 30 = 10 km/hr Step 2: Convert to m/s: 10 × (5/18) = 50/18 = 25/9 m/s Step 3: Total distance = 150 + 200 = 350 m Step 4: Time = 350 ÷ (25/9) = 350 × 9/25 = 126 seconds

Answer: 126 seconds = 2 minutes 6 seconds

Common Errors

• Forgetting to convert km/hr to m/s → Multiply by 5/18. Don’t mix units!
• Using sum instead of difference for same direction → Same direction = subtract speeds, opposite = add
• Forgetting train length in platform problems → Always add the train’s own length to platform length