JEE: Power with Variable Force 💡
❓ Concept Question How do we calculate instantaneous power when the force is time-dependent ? 🖼 Concept Image ✍️ Short Concept Power represents rate of doing work . For a moving particle: P = F ⃗ ⋅ v ⃗ P = \vec{F} \cdot \vec{v} If force varies with time, velocity must be found using Newton’s law and integration . 🔷 Step 1 — Instantaneous Power Definition 💯 Power is defined as: P = F ⃗ ⋅ v ⃗ P = \vec{F} \cdot \vec{v} This is a dot product . 👉 Only the component of force along velocity contributes to power. 🔷 Step 2 — When Force Depends on Time If: F ⃗ = F ( t ) \vec{F} = F(t) Then acceleration becomes: a ⃗ = F ⃗ m \vec{a} = \frac{\vec{F}}{m} Since velocity changes with time: v ⃗ = v ( t ) \vec{v} = v(t) 🔷 Step 3 — Finding Velocity Using Newton’s second law: F = m a F = ma and a = d v d t a = \frac{dv}{dt} So: d v d t = F ( t ) m \frac{dv}{dt} = \frac{F(t)}{m} Integrating: v = ∫ F ( t ) m d t v = \int \frac{F(t)}{m} \, dt 👉 Integration gives vel...