Question

Two ships $A$ and $B$ are $10km$ apart on a line running south to north. Ship $A$ farther north is streaming west at $20km/hr$ and ship $B$ is streaming north at $20km/hr$. What is their distance of closest approach and how long do they take to reach it?

• A. $4\sqrt{2}km,15min$
• B. $5\sqrt{5}km,15min$
• C. $5\sqrt{2}km,20min$
• D. $2\sqrt{2}km,15min$

Answer 1

The correct option is B $5\sqrt{5}km,15min$

Given,

Initial separation in both ships is $10km$.

Ship A stream west at speed $v_A=\frac{dx}{dt}=20km/s$ (ship A movement increase value of x).

Ship B stream west at speed $v_B=\frac{-dy}{dt}=20km/s$ (ship B movement decrease value of Y).

Distance between both ships is $s$

$s^2=x^2+y^2$

Differentiate with respect to time

$2sds=2xdx+2ydy$

For minimum, approach first differentiation is zero.

$2xdx+2ydy=0$

$\Rightarrow \frac{x}{y}=\frac{-dy}{dx}=\frac{-dy/dt}{dx/dt}=\frac{20}{20}=1$

$\Rightarrow x=y$

$\Rightarrow 20t=10-20t$

$\Rightarrow t=0.25hr=15min$

$x=20t=20\times 0.25=5km$

$x=y=5km$

$s=\sqrt{x^2+y^2}=\sqrt{5^2+5^5}=5\sqrt{5}km$

Minimum approach is $5\sqrt{5}km$, In time $15min$