Time, Speed & Distance

Speed, time and distance problems are built on one relationship — everything else (trains, boats, relative motion) is that relationship applied to a specific situation.

The core formula

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

Unit conversion: to convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Relative speed

Moving in the same direction: relative speed = difference of speeds. Moving in opposite directions: relative speed = sum of speeds.

Boats and streams

Downstream speed = boat speed + stream speed. Upstream speed = boat speed − stream speed.

Worked Example 1

An UdaanPath outreach team drives 180 km to a coaching-institute partner meeting in 3 hours. Find their speed in km/h and m/s.

Speed = 180/3 = 60 km/h. In m/s: 60 × 5/18 ≈ 16.67 m/s.

Worked Example 2 — trains

A train 150 m long crosses a pole in 10 seconds. Find its speed in km/h.

Speed = 150/10 = 15 m/s = 15 × 18/5 = 54 km/h.

Worked Example 3 — boats

A boat's speed in still water is 10 km/h and the stream flows at 2 km/h. Find the time to travel 24 km downstream.

Downstream speed = 10 + 2 = 12 km/h. Time = 24/12 = 2 hours.

Common mistakes

When a train "crosses" a platform (not just a pole), the distance covered equals the train's length plus the platform's length — forgetting to add the platform length is the most common error in this topic.

Practice Set

1. Convert 90 km/h to m/s.
2. Two trains, 120 m and 180 m long, run in opposite directions at 54 km/h and 36 km/h. Find the time they take to cross each other.
3. A boat covers 32 km upstream in 4 hours. Find the boat's speed upstream.
4. A man walks at 5 km/h and reaches his office 10 minutes late. Walking at 6 km/h, he reaches 5 minutes early. Find the distance to his office.