Key Formulas:
1. $\text{Distance = Speed × Time}$
2. $\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}$
3. $\text{Time} = \dfrac{\text{Distance}}{\text{Speed}}$
4. To convert kilometers per hour (kmph) into meters per second (m/s), multiply with $\dfrac{5}{{18}}$
5. To convert meters per second (m/s) into kilometers per hour (kmph), multiply with $\dfrac{18}{{5}}$
6. If the ratio of speeds is a : b : c, then the ratio of times taken is = $\displaystyle\frac{1}{a}:\frac{1}{b}:\frac{1}{c}$
Average Speed:
7. Average speed = $\displaystyle\frac{{{\text{Total distance travelled}}}}{{{\text{Total time taken}}}}$
8. A person travels from A to B and then returns to A at speeds $x$ and $y$, then the average speed of the entire journey = $\dfrac{{2xy}}{{x + y}}$
9. A person travels from P to Q and then goes from Q to R taking equal times with speeds $x$ and $y$, then the average speed of the entire journey = $\dfrac{{x + y}}{2}$
Relative speed Formulas:
Opposite Direction:
If two objects are moving in opposite directions towards each other at speeds $u$ and $v$, then Relative speed = Speed of first object + Speed of second object = $u + v$.
10. Time required for both objects to meet each other = $\dfrac{\text{Distance}}{\text{Relative speed}}$ = $\dfrac{D}{u + v}$
In the above case, time required for the boy to meet the girl = $\dfrac{\text{Distance}}{\text{Relative speed}}$ = $\dfrac{{80}}{{3 + 5}} = 10$ seconds.
Distance covered by the Girl by that time they meet each other = $5 m/s \times 10 sec$ = 50 meters
Distance covered by the Boy by that time they meet each other = $3 m/s \times 10 sec$ = 30 meters
Note:
Distances covered is directly proportional to speeds. So distance covered by the girl and boy should be equal to 5 : 3.
So, distance covered by the girl = $\dfrac{5}{{5 + 3}} \times 80 = 50$ meters
distance covered by the boy = $\dfrac{3}{{5 + 3}} \times 80 = 30$ meters
Same Direction:
If the two objects move in the same direction with speeds u and v, then Relative speed = difference between their speeds = $u - v$.
11. Time required for both objects to meet each other = $\dfrac{\text{Distance}}{\text{Relative speed}}$ = $\dfrac{D}{u - v}$
In the above case, time required for the police to catch the thief= $\dfrac{\text{Distance}}{\text{Relative speed}}$ = $\dfrac{{60}}{{8 - 5}} = 20$ seconds.
Distance covered by the police by that time he catches the thief = $8 m/s \times 20 sec$ = 160 meters
Distance covered by the thief by that time he was caught by the police = $6 m/s \times 20 sec$ = 120 meters
12. If the two objects start from A and B with speeds u and v respectively, and after crossing each other take a and b hours to reach B and A respectively, then u : v = $\sqrt {\displaystyle\frac{b}{a}} $
