time-speed-and-distance

Time and Distance Concepts

Concepts of time and distance with the conversion of units and their formulas are discussed below.

Speed

Speed of an object is the distance covered by it in a unit time interval.
\begin{align*}
Speed=\left[ \dfrac {Distance \ Travelled}{Time \ Taken}\right] \end{align*}
Units of measurement are km/h, m/s i, e kilometer per hour, meter per second respectively.

Conversion Of Units For Time And Distance Concepts

\begin{align*} One \ kilometer \ per \ hour = \dfrac {5}{18} \ m/\sec \end{align*}
\begin{align*} One \ meter \ per \ second= \dfrac {18}{5} \ km/\ hour
\end{align*}

Formulas of Time and Distance Concepts

1. \begin{align*}
Speed=\dfrac {Distance}{Time} \\ \\ (or) \\ \\ Time=\dfrac {Distance}{Speed} \\ \\ (or) \\ \\ Distance=Speed\times Time \end{align*}
2. \begin{align*} Average \ Speed=\dfrac {Total \ Distance}{Total \ Time}
\end{align*}
3. Let a man travels different distances d1,d2,d3………and so on in different times t1,t2,t3………. respectively then, \begin{align*}
Average \ Speed =\dfrac {Distance \ Travelled}{Time \ Taken} \\ \\ = \left[ \dfrac {\left( d_{1}+d_{2}+d_{3}+\ldots \right) }{\left( t_{1}+t_{2}+t_{3}+\ldots \right) }\right] km/h. \end{align*}
4. Let a man travels different distances d1,d2,d3………and so on with different speeds s1,s2,s3………. respectively then,
Average speed (km/h)\begin{align*} =\begin{bmatrix} d_{1}+d_{2}+d_{3}+\ldots \\ \overline {\left( \dfrac {d_{1}}{s_{1}}\right) +\left( \dfrac {d_{2}}{s_{2}}\right) +\left( \dfrac {d_{3}}{s_{3}}\right) }+\dots \end{bmatrix} \end{align*}
5. Let A goes from P to Q at a speed of s1 km/h and returns from Q to P at s2 km/hr speed then,
the average speed during the whole journey is \begin{align*} \left[ \dfrac {2S_{1}S_{2}}{S_{1}+S_{2}}\right]km/h. \end{align*}
7. Let A travels at a constant speed and covers distance d1 in t1 time and d2 distance in t2 time then,
\begin{align*} d_{1}\times t_{2}=d_{2}\times t_{1} \end{align*}
8. Let A increases its speed from X km/h to Y km/h to cover a distance in t2 hours in place of t1 hours then
Distance
\begin{align*}
\left[ \dfrac { Product \ of Speeds }{ Difference \ in \ Speeds }\right] \times \begin{bmatrix} Change \\ in \\ Time \end{bmatrix}\end{align*}
\begin{align*} =\left[ \dfrac {xy}{\left( x-y\right) }\right] \left( t_{2}-t_{1}\right)km. \end{align*}
9. Let A travels at a speed of s1 and reaches his destination t1 hours late but when he travels with a speed of s2 he reaches his destination t2 hours before then,
the distance between two places is \begin{align*} \left[ \dfrac {S_{1}\times S_{2}\times \left( t_{1}+t_{2}\right) }{\left( S_{2}-S_{1}\right) }\right]. \end{align*}
10. Let two persons  A and B starting at the same time from points P and Q towards each other, crosses each other at a point and they take t1 hours and t2 hours to reach the points Q and P respectively  then, \begin{align*}
\dfrac {A’s \ Speed}{B’s \ Speed}=\sqrt {\dfrac {t_{2}}{t_{1}}}.
\end{align*}
11. Let A’s new speed is x/y of the original speed, then
the change in time taken to cover the same distance \begin{align*} \left[ \left( \dfrac {b}{a}\right) -1\right] \times Original \ time. \end{align*}
12. Let A covers a 1/x part of the distance with speed s1, 1/y part with s2, and 1/z  part with s3 k/h and so on then,
the average speed to cover the whole distance is \begin{align*} \dfrac {1}{\left( \dfrac {1}{xs_{1}}\right) +\left( \dfrac {1}{ys_{2}}\right) +\left( \dfrac {1}{zs_{3}}\right) +\ldots }. \end{align*}

13. Relative speed

  • Let A and B run in opposite directions then, the relative speed is the sum of their speeds.
  • When A and B run in the same directions then relative speed is the Difference of their speeds.
  • Time taken by A and B to meet is
    \begin{align*} \dfrac {Distance \ between \ them}{Relative \ speed}. \end{align*}

14. Let A overtakes or follows B then
the time taken to catch B is \begin{align*} \dfrac {Distance \ between \ them}{Relative \ speed} \\ \\ (or) \\ \\ \left[ \dfrac {\left( speed \ of \ B \right) \times time}{Difference of speeds }\right]
\end{align*}
15. Let there is n number of poles which are at a distance of X meters apart from each other than, then
the distance between first and last pole is \begin{align*} \left( n-1\right) X.
\end{align*}

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