Question

A motor boat has a speed of 5m/s. At time t=0, it's position vector relative to a origin is (-11i^+16j^)m, havng the aim of getting as close as possible to a streamer. At time=0, the streamer is at the point (4i^+36j^)m and it is moving with a constant velocity (10i^-5j^)m/s. Find the direction in which the motor boat must steer

Answer 1

Answer:

North West

Explanation:

Answer 2

Concept:

To get as near to the steamer as feasible, the motorboat's motion should be perpendicular to the relative motion.

Given:

Speed of motor boat is 5m/s

Position vector of boat is -11î + 16ĵ

Position vector of streamer is 4î + 36ĵ

Velocity of streamer is 10î - 5ĵ

Find:

Direction in which motor boat should steer

Solution:

Let the velocity of boat be

Vb = aî + bĵ

Velocity of streamer is

Vs = 10î - 5ĵ

Relative velocity will be

Vbs = (a-10) î + (b+5) ĵ

For shortest distance Vb should be perpendicular to Vbs

Vb ⊥ Vbs means Vb . Vbs (Dot product) = 0

a(a-10) + b(b+5) = 0    ----1)

as speed of boat is 5 means

a²+ b² = 5²    ----2)

Solving equation 1 and 2 we get value of a = 4, b =3

Hence Velocity of boat will be Vb = 4 î + 3 ĵ

So the direction in which the motor boat must steer is 4 î + 3 ĵ .

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