First slide

Relative motion in 2 dimension

Question

Ship A is moving with velocity 30m/s due east and ship B with velocity 40m/s due north. Initial separation between the ships is 10km as shown in figure. After what time (in seconds) ships are closest to each other ?

Moderate

Solution

Velocity toward east, ${\vec{V}}_{A} = 30 \hat{i} m / s$
Velocity towards North ${\vec{V}}_{B} = 40 \hat{j} m / s$
Distance $A B = 10 km$
${\vec{V}}_{B A} = 40 \hat{j} - 30 \hat{i}$
$|{\vec{V}}_{B A}| = \sqrt{{(40)}^{2} + {(- 30)}^{2}} = 50 m / s$
$\Rightarrow \tan θ = \frac{4}{3}; |{\vec{V}}_{B A}| = 50 m / s$
Required time= $= \frac{A B \cos θ}{V_{B A}}$
$t = \frac{10^{4} \times \frac{3}{5}}{50} = \frac{3000}{25} = 120 s$

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