JEE Main: Find Power Using F · v Concept 💡

 

❓ Question

An object of mass 1000 g experiences a time dependent force

$$\vec F = (2t \,\hat i + 3t^2 \,\hat j)\, \text{N}$$

The power generated by the force at time t is:

---

🖼 Question Image

---

✍️ Short Solution

This is a Power = F · v question.

👉 Time dependent force
👉 Velocity changes with time
👉 First find velocity using Newton’s 2nd Law

Then apply:

$$P = \vec F \cdot \vec v$$

---

🔷 Step 1 — Convert Mass & Use Newton’s Law (MOST IMPORTANT 💯)

Given mass:

$$1000 \text{ g} = 1 \text{ kg}$$

Using:

$$\vec F = m\vec a$$

So acceleration:

$$\vec a = \vec F$$
$$\vec a = (2t \hat i + 3t^2 \hat j)$$

---

🔷 Step 2 — Find Velocity by Integration

Since:

$$\vec a = \frac{d\vec v}{dt}$$

Integrate:

x-component:

$$v_x = \int 2t\, dt = t^2$$

y-component:

$$v_y = \int 3t^2\, dt = t^3$$

So velocity:

$$\vec v = (t^2 \hat i + t^3 \hat j)$$

---

🔷 Step 3 — Apply Power Formula

Power:

$$P = \vec F \cdot \vec v$$
$$= (2t \hat i + 3t^2 \hat j) \cdot (t^2 \hat i + t^3 \hat j)$$

Dot product:

$$P = 2t \cdot t^2 + 3t^2 \cdot t^3$$
$$P = 2t^3 + 3t^5$$

---

🔷 Step 4 — Final Expression

$$P = 2t^3 + 3t^5$$

---

✅ Final Answer

$$\boxed{P = 2t^3 + 3t^5 \text{ W}}$$

---

⭐ Golden JEE Insight

Whenever:

$$\vec F = f(t)$$

Steps are ALWAYS:

1️⃣ Use $F = ma$
2️⃣ Integrate to get velocity
3️⃣ Apply $P = F \cdot v$

⚡ Power needs velocity — not acceleration!