Speed, Time, and Distance

1. Fundamental Concepts

• Speed: The rate at which an object covers a distance. It is usually measured in km/hr or m/s.
• Time: The duration taken to cover a particular distance.
• Distance: The total path covered by an object.

The basic relationship among these is given by:

Speed=

Rearranging, we also have:

Distance= Speed  Time

Time=

2. Unit Conversions

• km/hr to m/s: Multiply by  5/18
• m/s to km/hr: Multiply by  5/18

3. Relative Speed

• When two objects move in the same direction, their relative speed is the difference between their speeds.
• When two objects move in opposite directions, their relative speed is the sum of their speeds.

Example:

• If two trains are moving in the same direction with speeds S₁ and S₂ ,the relative speed is ∣S₁−S₂∣ .
• If they are moving in opposite directions, the relative speed is S₁+ S_2​.

4. Average Speed

• Average speed is defined as the total distance covered divided by the total time taken. It is particularly important when the speed changes during different segments of the journey.
• For a journey with two equal parts covered at speeds S₁ and S₂​, the average speed is given by:

Average Speed= 2  S₁  S₂ / S₁ + S₂

5. Time, Speed, and Distance in Different Scenarios

• Trains: Problems involving trains often include calculating the time taken to pass a stationary object (like a pole) or another moving object (like another train).
• Boats and Streams: Involves calculating effective speed when moving upstream (against the current) and downstream (with the current).
  • Upstream speed: S_upstream=S_boat−S_stream
  • Downstream speed: S_downstream=S_boat+S_stream
• Races: These problems may involve finding out the time advantage or distance advantage of one contestant over another.

6. Important Formulas

• Distance Covered: D=S×T
• Time Taken: T=D/S
• Speed: S=D/T ​
• Relative Speed:
  • Same Direction: S_relative=∣S₁−S₂∣
  •  Opposite Direction: S_relative=S₁+S₂

Average Speed (for a journey with two equal distances): S_average=2  S₁  S₂ / S₁ + S₂

 

Tips for Banking and SSC Exams:

• Focus on Speed: Time management is key; practice solving problems quickly and accurately.
• Practice Unit Conversion: Ensure you’re comfortable converting between different units.
• Understand the Scenarios: Familiarize yourself with typical scenarios like trains, boats, races, and their specific formulas.

Example 1: Basic Speed Calculation

Problem: A car covers a distance of 150 km in 3 hours. What is the speed of the car?

Solution:

Speed =  =  = 50 km/hr

Example 2: Time Taken to Cover a Distance

Problem: A cyclist travels at a speed of 12 km/hr. How long will it take to cover a distance of 54 km?

Solution:

Time =  =  = 4.5 hours

Example 3: Relative Speed (Same Direction)

Problem: Two trains are moving in the same direction. The first train moves at 60 km/hr, and the second train at 75 km/hr. What is their relative speed?

Solution:

Relative Speed=∣75 km/hr−60 km/hr∣=15 km/hr

Example 4: Relative Speed (Opposite Direction)

Problem: Two trains are moving towards each other. The first train moves at 80 km/hr, and the second train at 90 km/hr. What is their relative speed?

Solution:

Relative Speed=80 km/hr+90 km/hr=170 km/hr

Example 5: Average Speed

Problem: A man travels 120 km at 40 km/hr and then another 120 km at 60 km/hr. What is his average speed for the entire journey?

Solution:

Average Speed =

Example 6: Time Taken by a Train to Pass a Pole

Problem: A train 200 meters long is running at a speed of 72 km/hr. How much time will it take to pass a pole?

Solution:

• Convert speed to m/s:

72 km/hr=  m/s = 20 m/s

 

Time taken:

Time=  =  = 10 sec

Example 7: Boats and Streams (Upstream and Downstream)

Problem: A boat’s speed in still water is 8 km/hr, and the speed of the stream is 2 km/hr. How long will it take for the boat to travel 24 km upstream?

Solution:

• Speed upstream:

S_upstream=8 km/hr−2 km/hr=6 km/hr

• Time taken:

    T =  = 4 hrs

Example 8: Distance Covered by Two Moving Objects

Problem: Two trains start at the same time from stations 300 km apart and move towards each other. One train travels at 50 km/hr and the other at 70 km/hr. How long will it take for them to meet?

Solution:

• Combined speed (since they are moving towards each other):

S_combined=50 km/hr+70 km/hr=120 km/hr

 

Time taken:

T =  = 2.5 hrs

Example 9: Conversion and Distance Covered

Problem: A person jogs at a speed of 1.5 m/s. How many kilometers does he cover in 30 minutes?

Solution:

• Convert time to seconds:

30 minutes=30×60 seconds=1800 seconds

• Distance covered:

Distance=Speed×Time=1.5 m/s×1800 seconds=2700 meters