Minimum Distance Between Two Ships | Relative Motion in 2D | JEE Physics Doubt | Doubtify JEE

 

🚢 Minimum Distance Between Two Moving Ships – Relative Motion | JEE Physics | Doubtify JEE

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💡 Question:

Ship A is sailing towards the north-east with velocity
v = 30 î + 50 ĵ km/h,
where î points east and ĵ points north.

Ship B is located at a distance of 80 km east and 150 km north of ship A and is sailing towards the west at 10 km/h.

Question: After how much time will Ship A be at a minimum distance from Ship B?

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🖼️ Question Image:

🧠 Solution (Image):

✍️ Explanation:

To find the time at which the two ships are closest, we use relative motion. Let’s break down the scenario:

• Initial relative position of B w.r.t. A:

  • Δx = 80 km (east)

  • Δy = 150 km (north)

  • So, relative position vector:
    r₀ = 80 î + 150 ĵ km

• Velocity of A:
  vₐ = 30 î + 50 ĵ km/h

• Velocity of B:
  v_b = -10 î km/h (since it's moving west)

• Relative velocity (vᵣₑₗ = v_b - v_a):
  = (-10 - 30) î + (0 - 50) ĵ
  = -40 î - 50 ĵ km/h

Now, let the position of B relative to A at time t hours be:
r(t) = r₀ + vᵣₑₗ × t

To minimize the distance, we minimize the magnitude of r(t). This happens when r(t) is perpendicular to vᵣₑₗ.

So,

r(t) ⋅ vᵣₑₗ = 0

Substitute:

(80 + (-40)t)(-40) + (150 + (-50)t)(-50) = 0

Solving:

-3200 + 1600t - 7500 + 2500t = 0
4100t = 10700
t = 10700 / 4100 = 2.61 hours (approx.)

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🎥 Video Solution:

🔍 Why this Question is Important:

• Tests Relative Velocity in 2D, a key concept in Kinematics.

• Frequently appears in JEE Mains and Advanced papers.

• Enhances conceptual clarity in motion along different axes.

• Real-world application: used in navigation, aviation, and radar systems.

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🧠 Pro Tip:

In relative motion problems, always switch to the frame of one object. That simplifies your reference and reduces calculation errors.

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