What is the ratio of speed?

Question

If two trains start at same time from point A and B towards each other and after crossing they take |a| and |b| seconds in reaching B and A respectively.

What is the ratio of speed of A and B.

Answer 1

Let's suppose speed of $A$ and $B$ is $s_1$ and $s_2$. time taken to meet is $t$. and total distance between $A$ and $B$ is $x$. Then:

$s_1(t) + s_2(t)=x$ $......$equation $(1)$

$s_1(t+a) =x$ $......$equation$(2)$

$s_2(t+b)=x$ $......$equation$(3)$

From $(1)$ and $(2)$:

$s_1(t)+s_2(t)=s_1(t)+s_1(a)$

So: $s_2(t)=s_1(a)$

which implies that: $\frac{s_1}{s_2} = \frac{t}{a}$ $......$equation$(4)$

From $(1)$ and $(3)$:

$$s_1(t)+s_2(t)=s_2(t) +s_2(b)$$

So: $s_1(t)=s_2(b)$

which implies that: $\frac{s_1}{s_2}=\frac{b}{t}$ $......$equation$(5)$

So, from equations $(4)$ and $(5)$:

$$\frac{t}{a}=\frac{b}{t}\rightarrow t^2=b\times a\rightarrow t=\sqrt{ba}$$

Now we can easily find ratio by putting value of $t$ in equation $(2)$ and $(3)$:

$$\frac{s_1}{s_2}=\frac{\sqrt b}{\sqrt a}$$