Linear Tracks(Races)
If A and B both participate in an L metre race, and A beats B by x metres(say), then the time taken by A to complete L metres is the same as the time taken by B to complete (L-x) metres.
In that case, if speed of A is a m/s and B is b m/s then $$\dfrac{L}{a}=\dfrac{L-x}{b}$$
So, the ratio of their speeds is $$\dfrac{a}{b}=\dfrac{L}{L-x}$$
Beats by t seconds
If A and B both participate in an L metre race, and A beats B by t seconds, it means:
- Both A and B run the full L metres
- A finishes t seconds before B
So if speed of A is a m/s and speed of B is b m/s:
$$\frac{L}{b} - \frac{L}{a} = t$$
Meeting Time:
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
$$\frac{S_{1}}{S_{2}}$$=$$\sqrt{\frac{T_{2}}{T_{1}}}$$
