Kinetic Energy and Conservation of Energy, Part 1

I’ve been wanting to systematically go through a physics text, refreshing my knowledge and filling in gaps. I found The Feynman Lectures on Physics, freely available online, based on lectures from the great physicist and physics popularizer, Richard Feynman. As I read Chapter 4: “Conservation of Energy,” I wonder if I can derive the formula for kinetic energy, the energy an object has due to its motion. My primary motivation is that while it is relatively easy for me to grasp why the gravitational potential energy is U=mgh (that is, weight times height), I have a much harder time intuitively understanding why the kinetic energy should be proportional to the velocity squared: Kmv².

Problem: Given the formula for gravitational potential energy, can I use the principle of conservation of energy (of a projectile) to derive the equation for its kinetic energy?

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