Time and Distance

Speed, relative motion, average speed

Time, Speed and Distance is a fundamental topic in quantitative aptitude that deals with the relationship between three quantities: time taken, speed maintained, and distance covered. These problems appear frequently in competitive exams and require understanding of basic formulas, unit conversions, and various scenarios like relative speed, average speed, and meeting points.

Key Formulas

\text{Speed = Distance / Time}

\text{Distance = Speed x Time}

\text{Time = Distance / Speed}

\text{Average Speed = Total Distance / Total Time}

\text{Relative Speed (same direction) = |Speed - Speed |}

\text{Relative Speed (opposite direction) = Speed + Speed }

\text{Conversion: 1 km/hr = 5/18 m/s}

\text{Conversion: 1 m/s = 18/5 km/hr}

\text{If a person travels equal distances at speeds x and y, Average Speed = 2xy/(x+y)}

\text{If distance is constant, Speed 1/Time (inverse proportion)}

Key Concepts

Basic Relationships

The three fundamental quantities are related by Speed = Distance/Time. If any two are known, the third can be calculated. Always ensure units are consistent--convert km/hr to m/s when working with meters and seconds, or vice versa using the conversion factors 5/18 and 18/5.

Average Speed

Average speed is always Total Distance divided by Total Time, NOT the arithmetic mean of speeds. For equal distances traveled at different speeds, use the harmonic mean formula: 2xy/(x+y). For different distances, calculate total distance and total time separately.

Relative Speed

When two bodies move in the same direction, their relative speed is the difference of their speeds. When moving in opposite directions, their relative speed is the sum of their speeds. This concept is crucial for problems involving trains, races, and meeting points.

Trains and Platforms

A train crossing a stationary object (pole/person) covers distance equal to its own length. Crossing a platform or bridge requires covering the sum of train length and platform length. When two trains cross each other, the distance covered is the sum of both train lengths.

Boats and Streams

Upstream speed (against current) = Boat speed - Stream speed. Downstream speed (with current) = Boat speed + Stream speed. Still water speed = (Upstream + Downstream)/2. Stream speed = (Downstream - Upstream)/2.

Circular Tracks and Races

On a circular track, when two runners start together and run in the same direction, the faster gains one complete round every time the relative distance covered equals the track length. For opposite directions, they meet when the sum of distances equals the track length.

Solved Examples

Problem 1:

A car travels from city A to city B at 60 km/hr and returns at 40 km/hr. If the total time taken is 10 hours, find the distance between the cities.

Solution:

Let distance = D km.
Time going = D/60 hours.
Time returning = D/40 hours.
Total time = D/60 + D/40 = 10
Finding common denominator (120):
(2D + 3D)/120 = 10
5D/120 = 10
D = 10 x 120/5 = 240 km
Answer: The distance between cities is 240 km.

Problem 2:

Two trains of length 150m and 200m are running towards each other at 54 km/hr and 36 km/hr respectively. How long will they take to cross each other?

Solution:

Convert speeds to m/s:
Train 1: 54 x (5/18) = 15 m/s
Train 2: 36 x (5/18) = 10 m/s
Relative speed (opposite direction) = 15 + 10 = 25 m/s
Total distance to cover = 150 + 200 = 350m
Time = Distance/Speed = 350/25 = 14 seconds
Answer: They will cross each other in 14 seconds.

Problem 3:

A man can row 30 km upstream in 6 hours and 40 km downstream in 4 hours. Find the speed of the man in still water and the speed of the stream.

Solution:

Upstream speed = 30/6 = 5 km/hr
Downstream speed = 40/4 = 10 km/hr
Speed in still water = (Upstream + Downstream)/2 = (5 + 10)/2 = 7.5 km/hr
Speed of stream = (Downstream - Upstream)/2 = (10 - 5)/2 = 2.5 km/hr
Answer: Speed in still water is 7.5 km/hr, speed of stream is 2.5 km/hr.

Problem 4:

A person covers half of his journey at 30 km/hr and the remaining half at 60 km/hr. Find his average speed for the entire journey.

Solution:

For equal distances at different speeds, use: Average Speed = 2xy/(x+y)
Where x = 30 km/hr and y = 60 km/hr
Average Speed = (2 x 30 x 60)/(30 + 60) = 3600/90 = 40 km/hr
Answer: The average speed is 40 km/hr.

Tips & Tricks

Always verify unit consistency before applying formulas--convert km/hr to m/s by multiplying by 5/18 when working with meter distances.
For average speed problems, remember it is total distance divided by total time, not the arithmetic mean of individual speeds.
In train problems, carefully identify what the train is crossing--a pole (just train length), a platform (train + platform), or another train (sum of both lengths).
When two objects move towards each other, add their speeds for relative speed. When moving in the same direction, subtract the slower from the faster.
For problems involving stoppages, calculate the running time and stopped time separately, then use total time for average speed calculations.
Practice converting between different time units (hours to minutes to seconds) and distance units (km to m) to avoid calculation errors under exam pressure.

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