Anecdotes please - quadratic equations in real life

[Albert Chan]

(05-05-2024 11:42 PM)Thomas Klemm Wrote:  \(
\begin{align}
t &= \frac{s_1}{v - u_1} + \frac{s_2}{v + u_2} \\
25 &= \frac{300}{v - 35} + \frac{450}{v + 40} \\
\end{align}
\)
...

Therefore my speed is \(v = 50\text{m}/\text{min}\).
The other solution is \(-25\text{m}/\text{min}\) and doesn't make sense.

It may not be obvious why negative velocity is wrong.
If we play the film backward, even time can be negative.

However, time cannot be positive and negative at the same time.
RHS time terms must have the same sign: (v > 35) or (v < -40)
This is why v = -25 is wrong.

Another way, |x| < 0.5, to ensure both time terms have same sign.

1 = 12/(v-35) + 18/(v+40) = (.5+x) + (.5-x)
v = 12/(.5+x)+35 = 18/(.5-x)-40

18/(.5-x) - 12/(.5+x) = 75
.24/(.5-x) - .16/(.5+x) = 1
(.24-.16)*.5 + (.24+.16)*x = .5^2 - x^2

x^2 + .4*x - .21 = 0
x = -.2 ± √(.2^2 + .21) = -.2 ± .5 = .3, -.7
v = 12/(.5+x)+35 = 50

Or, we can just iterate for solution, without quadratic formula
This may be faster than getting quadratic coefficients.

v = 35 + 12 / (1 - 18/(v+40))
v: 35+12=47 → 50.130 → 49.994 → 50.000