Time, Speed and Distance

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Theory

The questions in this section can vary from being very easy to surprisingly difficult. This is a conceptual section (especially questions involving clocks) and some of the questions can consume a lot of time. While solving, write down the equations as far as possible to avoid mistakes. The few extra seconds can help you avoid silly mistakes. Also, check if the units of distance, speed and time match up. So if you see yourself adding a unit of distance like m to a unit of speed m/s, you would realize you have missed a term. Choose to apply the concept of relative speed wherever possible as it can greatly reduce the complexity of the problem. Like speed and distance, in time and work while working with terms ensure that you convert all terms to consistent units like man-hours.

Theory

Constant Distance:

Let the distances travelled in each part of the journey be $$d_{1},  d_{2}, d_{3}$$ and so on till $$d_{n}$$ and the speeds in each part be $$s_{1},  s_{2},  s_{3}$$ and so on till $$s_{n}$$.

If $$d_{1} = d_{2} = d_{3} =...= d_{n}$$= d, then the average speed is the harmonic mean of the speeds $$s_{1}, s_{2}, s_{3}$$ and so on till $$s_{n}$$.

Constant Time:

Let the distances travelled in each part of the journey be $$d_{1},  d_{2}, d_{3}$$ and so on till $$d_{n}$$ and the time taken for each part be $$t_{1}, t_{2}, t_{3}$$ and so on till $$t_{n}$$.

If $$t_{1} = t_{2} = t_{3} =...= t_{n}$$= t, then the average speed is the arithmetic mean of the speeds $$s_{1}, s_{2}, s_{3}$$ and so on till $$s_{n}$$.

Tip
  • In a journey travelled with different speeds, if the distance covered in each stage is constant, the average speed is the harmonic mean of the different speeds.
  • In a journey travelled with different speeds, if the time travelled in each stage is constant, the average speed is the arithmetic mean of the different speeds.
Formula Questions

CLOCKS:

  • In a well functioning clock, both hands meet after every 720 / 11 Mins.
  • It is because the relative speed of the minute hand with respect to the hour hand = 11/2 degrees per minute.
Formula Questions

Circular Tracks

If two people are running on a circular track with speeds in the ratio a:b where a and b are co-prime, then

  • They will meet at a+b distinct points if they are running in the opposite directions.
  • They will meet at |a-b| distinct points if they are running in the same direction.

If two people are running on a circular track having perimeter I, with speeds m and n,

  • The time for their first meeting = $$\frac{I}{(m+n)}$$

(when they are running in opposite directions)

  • The time for their first meeting = $$\frac{I}{(|m-n|)}$$

(when they are running in the same direction)

If a person P starts from A and heads towards B and another person Q starts from B and heads towards A and they meet after a time 't' then,

                                                        t = $$\sqrt{x*y}$$

where x = time taken (after the meeting) by P to reach B and y = time taken (after the meeting) by Q to reach A.

A and B started st a time towards each other. After crossing each other, they took $$T_{1}$$ hrs, $$T_{2}$$ hrs respectively to reach their destinations. If they travel at constant speeds $$S_{1}$$ and $$S_{2}$$ respectively all over the journey, Then
                                                     $$\frac{S_{1}}{S_{2}}$$=$$\sqrt{\frac{T_{2}}{T_{1}}}$$

Formula Questions

BOATS & STREAMS

  • If the speed of water is 'W' and speed of a boat in still water is 'B'

    1. Speed of the boat downstream is B+W
    2. Speed of the boat upstream is B-W

The direction along the stream is called downstream. And, the direction against the stream is called upstream.

If the speed of the boat downstream is x km/hr and the speed of the boat upstream is y km/hr, then

                                      Speed of boat in still water= $$\dfrac{x+y}{2}$$ km/hr
                                      Speed of stream= $$\dfrac{x-y}{2}$$ km/hr

    If the speed a person is x m/s, and the speed of the escalator is y m/s, then the relative speed is (x+y) m/s (if in same direction).

    If they are moving opposite direction, then the relative speed is (x-y) m/s.

      Formula Questions

      Distance = Speed$$\times$$Time

      Speed = $$\frac{Distance}{Time}$$

      Time = $$\frac{Distance}{Speed}$$

      While covering the Speed in m/s to km/hr. multiply it by 3.6. It is because 1m/s = 3.6 km/hr

      If the ratio of the speeds of A and B is a : b, then

      • The ratio of the times taken to cover the same distance is 1/a : 1/b or b : a.
      • The ratio of distance travelled in equal time intervals is a : b

                                             Average speed= $$\frac{Total Distance Travelled}{Total Time Taken}$$

      If a part of a journey is travelled at speed $$S_{1}$$ km/hr in $$T_{1}$$ hours and the remaining part at speed $$S_{2}$$ km/hr in $$T_{2}$$ hours then.
                                             Total distance travelled= $$S_{1}T_{1}$$+$$S_{2}T_{2}$$ km
                                                            Average speed=$$\frac{S_{1}T_{1}+S_{2}T_{2}}{T_{1}+T_{2}}$$ km/hr

      If $$D_{1}$$ km is travelled at speed of $$S_{1}$$ km/hr, and $$D_{2}$$ km is travelled at speed of $$S_{2}$$ km/hr then

                                                             Average Speed= $$\frac{D_{1}+D_{2}}{\frac{D_{1}}{S_{1}}+\frac{D_{2}}{S_{2}}}$$ km/hr

      • In a journey travelled at different speeds, if the distance covered in each stage is constant, the average speed is the harmonic mean of the different speeds.
      • Suppose a man covers a certain distance st x km/hr and an equal distance at y km/hr

      Then the average speed during the whole journey is $$\frac{2xy}{x+y}$$ km/hr

      • In a journey travelled with different speeds, if the time travelled in each stage is constant, the average speed is the harmonic mean of the different speeds.
      • If a man travelled for a certain time at the speed of x km/hr and travelled for an equal amount of time at the speed of y km/hr then

      Then the average speed during the whole journey is $$\frac{x+y}{2}$$ km/hr

      Formula Questions

      TRAINS:

      Two trains of length $$L_1$$ and $$L_2$$ travelling at speeds of $$S_1$$ and $$S_2$$ cross each other in

      • $$\frac{L_1+L_2}{S_1+S_2}$$ if they are going in opposite directions

      • $$\frac{L_1+L_2}{S_1-S_2}$$ if they are going in the same direction
      Formula Questions
      • While converting the speed in m/s to km/hr, multiply it by 3.6. It is because 1 m/s = 3.6 km/hr
      Tip
      • In a well functioning clock, both the hands meet after every $$\frac{720}{11}$$ mins. It is because relative speed of minute hand with respect to hour hand = $$\frac{11}{2}$$ degrees per minute. 
      Formula Questions

      Circular Tracks

      If two people are running on a circular track with speeds in the ratio a:b where a and b are co-prime, then

      • They will meet at a+b distinct points if they are running in the opposite directions.
      • They will meet at |a-b| distinct points if they are running in the same direction.

      If two people are running on a circular track having perimeter I, with speeds m and n,

      • The time for their first meeting = $$\frac{I}{(m+n)}$$

      (when they are running in opposite directions)

      • The time for their first meeting = $$\frac{I}{(|m-n|)}$$

      (when they are running in the same direction)

      Formula Questions

      If the speed is increased by $$\frac{a}{n}$$, the time will reduce by $$\frac{a}{n+a}$$  and vice versa.

      Or If the speed is decreased by $$\frac{a}{n}$$, the time will increase by $$\frac{a}{n-a}$$ and vice versa

      Because the Product of speed and time is distance, which remains constant. Hence, if one of the speeds or times increases, the other has to decrease.


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