SATHEE: Time, Speed & Distance - All Formulas & Shortcuts

Time, Speed & Distance - All Formulas & Shortcuts

Quick reference guide for all TSD formulas including trains, boats, and races

📘 Basic Formulas

1. Fundamental Relationship

Distance = Speed × Time

Speed = Distance / Time

Time = Distance / Speed

Units:

Speed: km/hr, m/s, miles/hr
Distance: km, m, miles
Time: hours, minutes, seconds

2. Unit Conversions ⭐⭐⭐

km/hr to m/s:

km/hr × 5/18 = m/s

m/s to km/hr:

m/s × 18/5 = km/hr

Quick Examples:

36 km/hr = 36 × 5/18 = 10 m/s
72 km/hr = 72 × 5/18 = 20 m/s
10 m/s = 10 × 18/5 = 36 km/hr
15 m/s = 15 × 18/5 = 54 km/hr

Memory Trick: “5/18 to go down (km→m), 18/5 to go up (m→km)”

3. Average Speed

For same distance at different speeds:

Average Speed = 2ab/(a+b) (Harmonic Mean)

Example: Going at 30 km/hr, returning at 20 km/hr

Avg = (2×30×20)/(30+20) = 1200/50 = 24 km/hr

Important: Average ≠ (30+20)/2 = 25!

For different distances:

Average Speed = Total Distance / Total Time

4. Relative Speed

Same direction:

Relative Speed = |S₁ - S₂|

Opposite direction:

Relative Speed = S₁ + S₂

Example:

A: 40 km/hr, B: 30 km/hr
Same direction: 40 - 30 = 10 km/hr
Opposite: 40 + 30 = 70 km/hr

🚂 Trains

5. Train Passing Objects

Train passing a pole/person:

Time = Length of train / Speed of train

Train passing a platform/bridge:

Time = (Length of train + Length of platform) / Speed

Example: 100m train at 20 m/s passes 150m platform

Time = (100+150)/20 = 250/20 = 12.5 seconds

6. Two Trains

Same direction:

Time to cross = (L₁ + L₂) / (S₁ - S₂)

Opposite direction:

Time to cross = (L₁ + L₂) / (S₁ + S₂)

Example: 120m train at 60 km/hr, 80m train at 40 km/hr (opposite)

Combined length = 200m = 0.2 km Relative speed = 60 + 40 = 100 km/hr Time = 0.2/100 = 0.002 hr = 7.2 seconds

🚤 Boats & Streams

7. Boat in Still Water

Speed in still water = (Downstream + Upstream) / 2 Speed of stream = (Downstream - Upstream) / 2

Alternative:

Downstream = Still water + Stream Upstream = Still water - Stream

Example:

Downstream: 20 km/hr
Upstream: 10 km/hr
Still water = (20+10)/2 = 15 km/hr
Stream = (20-10)/2 = 5 km/hr

8. Time Calculations

Going downstream, returning upstream:

Total time = D/Downstream + D/Upstream

Average speed:

Avg = (Downstream × Upstream) / (Still water) OR Avg = (D² - S²) / D (where D = downstream, S = stream)

🏃 Races

9. Dead Heat Race

A beats B by ‘x’ meters:

When A finishes race, B is ‘x’ meters behind

Ratio of speeds:

A:B = Race distance : (Race distance - x)

Example: 100m race, A beats B by 10m

Speed ratio = 100:90 = 10:9

10. Time-based Race

A beats B by ’t’ seconds:

When A finishes, B takes ’t’ more seconds

Ratio of speeds:

A:B = Time_B : Time_A = (T+t) : T

11. Head Start

A gives B a start of ‘x’ meters:

A runs full distance B runs (distance - x)

For dead heat:

Speed_A × T = Distance Speed_B × T = Distance - x

⚡ Quick Shortcuts

12. Distance = Speed × Time Variations

Common patterns:

Speed	Time	Distance
60 km/hr	2 hr	120 km
80 km/hr	3 hr	240 km
50 km/hr	4 hr	200 km

Mental math: For 60 km/hr, multiply hours by 60

13. Meeting Point

A and B start from opposite ends toward each other:

Time to meet = Distance / (Speed_A + Speed_B)

Meeting point from A = Speed_A × Time Meeting point from B = Speed_B × Time

Ratio of distances:

Distance_A : Distance_B = Speed_A : Speed_B

14. Catch-up Problems

A chases B (B has head start):

Time to catch = Head start / (Speed_A - Speed_B)

Example: B is 10 km ahead. A: 50 km/hr, B: 30 km/hr

Time = 10/(50-30) = 10/20 = 0.5 hr = 30 min

🔥 Advanced Formulas

15. Escalator/Moving Walkway

Person’s effective speed = Own speed ± Escalator speed

Going up: Add Going down: Subtract (or add if faster than escalator)

Time_up = Length / (Speed_person + Speed_escalator) Time_down = Length / (Speed_person - Speed_escalator)

16. Circular Track

Meeting point (opposite directions):

Time = Circumference / (S₁ + S₂)

Meeting point (same direction):

Time = Circumference / |S₁ - S₂|

After meeting, when will they meet again?

Same calculation as first meeting

17. Journey with Rest

Total time includes rest:

Total time = Travel time + Rest time Travel time = Distance / Speed

Effective speed (including rest):

Effective = Distance / Total time

💡 Mental Math Techniques

Trick 1: Quick km/hr to m/s

For multiples of 18:

18 km/hr = 5 m/s 36 km/hr = 10 m/s 54 km/hr = 15 m/s 72 km/hr = 20 m/s 90 km/hr = 25 m/s

For others: Use 5/18 multiplier

Trick 2: Average Speed Shortcut

Two equal distances at ‘a’ and ‘b’ km/hr:

Instead of: 2ab/(a+b)

Quick: If a=30, b=20

Product: 30×20 = 600 Sum: 30+20 = 50 Double: 2×600 = 1200 Divide: 1200/50 = 24

Trick 3: Train Length from Time

Train crosses pole in ’t’ seconds at ’s’ m/s:

Length = s × t meters

Example: 10 sec at 15 m/s

Length = 150 meters

📊 Common Exam Patterns

Pattern 1: Speed Increased/Decreased

Speed increased by x%:

New time = Original time × 100/(100+x) Time saved = Original time × x/(100+x)

Speed decreased by x%:

New time = Original time × 100/(100-x) Extra time = Original time × x/(100-x)

Pattern 2: Time Saved/Lost

Late by ’t’ hours at speed ‘s₁’, on time at speed ‘s₂’:

Distance = t × s₁ × s₂ / (s₂ - s₁)

Pattern 3: Part of Journey

Distance D, x km at speed s₁, remaining at s₂:

Total time = x/s₁ + (D-x)/s₂

🎯 Quick Reference Table

Speed Conversions

km/hr	m/s
18	5
36	10
54	15
72	20
90	25
108	30
126	35
144	40

Pattern: Divide by 3.6 OR multiply by 5/18

Train Problems Checklist

✅ Convert speed to m/s (if needed) ✅ Add lengths for crossing ✅ Check direction (add for opposite, subtract for same) ✅ Time = Total distance / Relative speed

Boat Problems Checklist

✅ Identify downstream/upstream ✅ Calculate still water and stream speed ✅ Use: Down = Still + Stream, Up = Still - Stream ✅ For time: Distance/Speed

💎 Golden Rules

Always convert units to same system (either km/hr or m/s)
For average speed: Use harmonic mean for same distance
Trains: Add lengths when crossing each other
Boats: Downstream is always faster than upstream
Races: Speed ratio = Inverse of time ratio

🔍 Common Mistakes

❌ Using arithmetic mean instead of harmonic mean for average speed ❌ Forgetting to add train lengths ❌ Not converting km/hr to m/s for train problems ❌ Adding speeds in same direction (should subtract!) ✅ Always sketch diagram for complex problems ✅ Check if answer makes logical sense

Practice Questions: