Find Direction of a Moving Mosquito in 3D – Motion in a Plane JEE PYQ

 

  🧲 Motion in a Plane | Physics | Doubtify JEE

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💡 Question:

A mosquito is moving with a velocity v = 0.5 t² î + 3t ĵ + 9 k̂ m/s and accelerating in uniform conditions.What will be the direction of motion of the mosquito after 2 seconds?

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🖼️ Question Image:

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🔬 Concept Overview:

This is a vector motion problem where you're given a time-dependent velocity vector in three dimensions. To find the direction of motion, we must compute the velocity vector at $t = 2$ seconds and then determine its direction using unit vector analysis or angle calculation with respect to axes. This is a key concept from the chapter "Motion in a Plane", often asked in JEE Mains or Advanced.

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🧠 Step-by-Step Explanation:

Given:

$$\vec{v}(t) = 0.5t^2 \hat{i} + 3t \hat{j} + 9 \hat{k}$$

At $t = 2$ seconds:

• i-component: $0.5\times 2^2=2\text{m/s}$

• j-component: $3 \times 2 = 6 \, \text{m/s}$
  

• k-component: $9 \, \text{m/s}$
  

So,

$$\vec{v}(2) = 2\hat{i} + 6\hat{j} + 9\hat{k}$$

This vector gives us the instantaneous velocity at 2 seconds.

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🧭 Finding the Direction:

To find the direction, you can write the unit vector in the direction of velocity:

$$|\vec{v}| = \sqrt{2^2 + 6^2 + 9^2} = \sqrt{4 + 36 + 81} = \sqrt{121} = 11$$

So, the unit vector in the direction of motion:

$$\hat{v} = \frac{2}{11} \hat{i} + \frac{6}{11} \hat{j} + \frac{9}{11} \hat{k}$$

Thus, the mosquito is moving in a 3D direction with the above vector components at 2 seconds. You can also find angles with respect to axes using:

$$\cos(\theta_x) = \frac{2}{11}, \quad \cos(\theta_y) = \frac{6}{11}, \quad \cos(\theta_z) = \frac{9}{11}$$

This means the direction is closer to the z-axis, moderately inclined to the y-axis, and least toward the x-axis.

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✅ Final Answer:

Direction of motion at t = 2 seconds is along the vector:

$$2\hat{i} + 6\hat{j} + 9\hat{k}$$

or, as a unit vector:

$$\frac{2}{11} \hat{i} + \frac{6}{11} \hat{j} + \frac{9}{11} \hat{k}$$

🧠 Why this Question is Important:

This is a typical JEE Mains-style vector question that tests your grip on resolving velocity vectors and applying them to real-world motion analysis. These questions enhance your visualization of motion in 3D space, which becomes crucial when solving projectile motion or electromagnetism problems in later chapters.

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🧠 Solution (Image):

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🎥 Video Solution:

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