Dependence of mass on speed. Relativistic dynamics. The law relating mass and energy
Newton's laws fail as speed approaches c: however long a force acts on a particle, its speed never reaches c. To describe this, relativistic dynamics takes the mass of a moving body to be m = m₀/√(1 − υ²/c²) (m₀ is the rest mass), momentum p = mυ and force F = Δp/Δt. At small speeds m ≈ m₀. The total energy of a body is E = mc², its rest energy is E₀ = m₀c², and its kinetic energy is E_k = E − E₀ = (m − m₀)c². So if the energy changes, the mass changes too: Δm = ΔE/c²; mass and energy are two aspects of one thing. Energy and momentum are connected by E² = (m₀c²)² + (pc)²; if m₀ = 0 then E = pc, and such a particle (for example the photon) moves at exactly c in every frame. Note: in modern physics «mass» often means only m₀, and the speed dependence is expressed through E and p; here we follow the textbook terminology.
In your notebook use E_k/E₀ = m/m₀ − 1 to compute E_k/E₀ for β = 0.6, 0.8, 0.9 and check numerically that at small speeds it turns into E_k ≈ m₀υ²/2 (take β = 0.1). Calculation only.