JET: English-Lecture-III
Time & Distance
QUIZ
Time & Distance is one of the most fundamental chapters in aptitude tests. All questions revolve around three core quantities:
Distance (D), Speed (S), and Time (T).
These three are connected by the golden formula:
\[\textbf{Speed} = \frac{Distance}{Time}, \quad \textbf{Time} = \frac{Distance}{Speed}, \quad \textbf{Distance} = Speed \times Time\]1. Basic Formula Applications
(A) Constant Speed Problems
When speed remains the same, distance is directly proportional to time.
Example:
A bike moves at $40\text{ km/h}$. Distance covered in 3 hours:
\(40 \times 3 = 120\text{ km}\)
(B) Unit Conversion Essentials
- $1\text{ km/h} = \frac{5}{18} \text{ m/s}$
- $1\text{ m/s} = \frac{18}{5} \text{ km/h}$
These conversions are frequently required in exams.
2. Average Speed
Average speed is not the simple average of speeds unless equal time is spent.
Formula
\(\textbf{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\)
Special Case: Same Distance
If a vehicle travels the same distance at speeds $a$ and $b$:
\[\textbf{Average Speed} = \frac{2ab}{a + b}\]Example
A car travels 60 km at 30 km/h and returns 60 km at 60 km/h:
\[\text{Avg speed} = \frac{2 \times 30 \times 60}{30 + 60} = 40 \text{ km/h}\]3. Relative Speed
Relative speed comes into play when two objects move towards, away, or in the same direction.
(A) Opposite Directions
\(\text{Relative Speed} = S_1 + S_2\)
(B) Same Direction
\(\text{Relative Speed} = |S_1 - S_2|\)
Example
Two trains of 40 km/h and 60 km/h moving in opposite directions:
Relative speed = 100 km/h
4. Trains and Platforms
Trains have length, so distance covered = sum of lengths involved.
Formula
\(\text{Time} = \frac{\text{Length of Train}}{\text{Speed}}\)
Common Cases
- Train passing a pole
Distance = length of train - Train passing a platform of length L
Distance = train length + platform length - Two trains crossing each other
Distance = sum of lengths
5. Boats and Streams
Flowing water affects speed.
Definitions
- Still water speed = $u$
- Stream speed = $v$
\(\textbf{Downstream speed} = u + v\) \(\textbf{Upstream speed} = u - v\)
Example
Boat speed = 10 km/h; stream = 2 km/h
Downstream → 12 km/h
Upstream → 8 km/h
6. Races & Circular Tracks
(A) Races
If A beats B by distance d:
- A covers full distance
- B covers (Distance – d)
(B) Circular Track (Meeting Point)
When two people start from the same point:
\[\text{Time to meet} = \frac{\text{Track Length}}{\text{Relative Speed}}\]Solved Examples
(A) Basic
A car travels 150 km at 50 km/h. Time taken: \(T = \frac{150}{50} = 3\text{ hours}\)
(B) Relative Speed
Two men run at 6 m/s and 8 m/s in the same direction on a track.
Relative speed = $8 - 6 = 2\text{ m/s}$
(C) Train Problem
A 300 m long train crosses a pole in 20 seconds.
Speed = $300/20 = 15\text{ m/s}$
Convert to km/h:
\(15 \times \frac{18}{5} = 54\text{ km/h}\)
(D) Boat Problem
Downstream time for 36 km if $u = 12, v = 3$:
Speed = $15\text{ km/h}$