derive, using the equations of motion, the formula for kinetic energy EK = 21mv2

What is Kinetic Energy?

Kinetic energy (KE) is the energy an object has because of its motion. Think of a ⚡️ rolling ball – the faster it rolls, the more energy it carries. The formula we’ll derive is $$E_K = \frac{1}{2}mv^2$$, where m is mass and v is speed.

We’ll use three simple ideas:

1. Newton’s Second Law: F = ma – force equals mass times acceleration.
2. Work Done: W = ∫F\,dx – the integral of force over distance.
3. Relationship between acceleration and velocity: a = dv/dt and dx = v\,dt.

Let’s combine them step by step:

1. Start with work: $$W = \int F\,dx$$.
2. Replace F with ma: $$W = \int m a\,dx$$.
3. Express a and dx in terms of velocity: a = dv/dt, dx = v\,dt → $$W = \int m \frac{dv}{dt} \, v\,dt$$.
4. Cancel the dt terms: $$W = \int m v\,dv$$.
5. Integrate with respect to v (assuming mass is constant): $$W = m \int v\,dv = m \left[\frac{v^2}{2}\right] = \frac{1}{2} m v^2$$.

Since the work done on an object increases its kinetic energy, we write $$E_K = \frac{1}{2} m v^2$$. 🎯

A 0.5 kg toy car speeds up to 4 m/s. What is its kinetic energy?

Plug into the formula:

$$E_K = \frac{1}{2} \times 0.5 \times (4)^2 = 0.25 \times 16 = 4 \text{ J}$$

So the car has 4 joules of kinetic energy. ⚡️

• Always show the full derivation – teachers love seeing the steps.
• Remember the units: kg · m² / s² = J (joule).
• When a problem asks for kinetic energy, check if you need to consider v² or v only.
• Use the work–energy theorem as a quick check: W = ΔE_K.
• Practice converting between F = ma, W = F·d, and KE = ½mv² – they’re all linked.