Direction of Motion from Velocity Vector at Given Time

❓ Question

A mosquito is moving with a velocity

$$\vec v=0.5t^2\hat i+3t\hat j+9\hat k \ \text{m/s}$$

and accelerating in uniform conditions. What will be the direction of motion of mosquito after $2$ s?

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🖼 Question Image

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✍️ Short Explanation

This is a vector motion problem.

👉 Direction of motion comes from velocity vector
👉 First calculate velocity at given time
👉 Then find direction ratios / unit vector.

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🔷 Step 1 — Velocity Vector at $t=2$ 💯

Given:

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

Substitute:

$$t=2$$
$$\vec v=0.5(2)^2\hat i+3(2)\hat j+9\hat k$$
$$\vec v=2\hat i+6\hat j+9\hat k$$

So direction ratios are:

$$2:6:9$$

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🔷 Step 2 — Magnitude of Velocity

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

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🔷 Step 3 — Direction Vector

Unit vector along motion:

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

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🔷 Step 4 — Direction Cosines

Direction cosines are:

$$\left(\frac2{11},\frac6{11},\frac9{11}\right)$$

So direction of motion is along:

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

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🔷 Step 5 — JEE Trap Alert 🚨

❌ Acceleration calculate karne lag jaana unnecessarily

❌ Velocity vector mein $t=2$ substitute bhool jaana

❌ Magnitude galat calculate kar dena

Remember:

$$\text{Direction of motion} \Rightarrow \text{Direction of velocity vector}$$

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✅ Final Answer

Direction of motion:

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

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