Learn Time, Speed, and Distance: Key Concepts & Formulas

Time, Speed, and Distance (TSD) is a crucial topic in quantitative aptitude, commonly tested in competitive exams and campus placements. This article simplifies TSD concepts, provides essential formulas, shortcuts, and practical examples, and suggests visualization techniques to enhance understanding.

Time, Speed, Distance Basics

Formulae:

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

Units of Measurement:

Speed: meters per second (m/s) or kilometers per hour (km/h)
Distance: meters (m) or kilometers (km)
Time: seconds (s), minutes (min), or hours (h)

Conversion Factors:

From m/s to km/h: Multiply by 18/5
From km/h to m/s: Multiply by 5/18

Pro Tip: Speed expressed in km/h is numerically larger than in m/s, helping you quickly identify correct conversions.

Relationships in TSD

Constant Speed: Distances are proportional to times.
Constant Time: Distances are proportional to speeds.
Constant Distance: Times are inversely proportional to speeds.

Key Formulas

a) Average Speed

When traveling at different speeds over equal distances:

$Average Speed=2uvu+v (where u and v are speeds).\text{Average Speed} = \frac{2uv}{u + v} \text{ (where } u \text{ and } v \text{ are speeds).}$

b) Relative Speed

Moving in the same direction: Relative speed = |u – v|
Moving in opposite directions: Relative speed = u + v

Types of TSD Problems

1) Ratio-Based Problems

If speeds are in the ratio $a:ba:b$ :

Distances covered in the same time = $a:ba:b$
Time taken to cover the same distance = $b:ab:a$

2) Overtaking or Passing

When a train passes a stationary object (e.g., pole or tree): Distance = Length of the train
When a train passes a platform: Distance = Length of train + Length of platform

3) Travel and Meeting

When two objects move towards each other, the time to meet is calculated as:

$Time to Meet=DistanceRelative Speed\text{Time to Meet} = \frac{\text{Distance}}{\text{Relative Speed}}$

4) Boats and Streams

Upstream speed = Boat speed – Stream speed
Downstream speed = Boat speed + Stream speed

From upstream (u) and downstream (v) speeds:

Speed in still water = $(u+v)/2(u + v) / 2$
Speed of the stream = $(v-u)/2(v - u) / 2$

5) Races

If one racer finishes ahead by $xx$ meters or $tt$ seconds, consider relative distances or times.
For circular tracks, use LCM to calculate when racers meet again at the starting point.

Sample Problems

1) Speed Conversion

Question: A person crosses a 400 m street in 6 minutes. What is their speed in km/h?

Solution:

Speed = $4006×60\frac{400}{6 \times 60}$ m/s = 1.11 m/s
Convert to km/h: $1.11×185=41.11 \times \frac{18}{5} = 4$ km/h

2) Time Adjustment

Question: A train covers 200 km in 4 hours. To cover the same distance in 2 hours, what speed is required?

Solution:

Distance = $200×4=800200 \times 4 = 800$ km
Speed = $8002=400\frac{800}{2} = 400$ km/h

3) Proportional Speeds

Question: Two trains run at speeds in a ratio of 6:8. If the second train travels 300 km in 3 hours, what is the speed of the first train?

Solution:

Speed of the second train: $3003=100\frac{300}{3} = 100$ km/h
Speed of the first train: $100×68=75100 \times \frac{6}{8} = 75$ km/h