E = m c^2 vs p = γ m v γ = 1 / sqrt(1 - v^2/c^2) F = dp/dt W = ∫ F dx = ∫ (dp/dt) dx = ∫ (dp/dt) (dx/dt) dt = ∫ v dp ∫ v dp = p v - ∫ p dv W = [p v]_{0→v} - ∫_{0}^{v} p dv = [ (γ m v) v ]_{0→v} - ∫_{0}^{v} (γ m v) dv = γ m v^2 - m ∫_{0}^{v} v (1 - v^2/c^2)^(-1/2) dv Let u = 1 - v^2/c^2 ⇒ du = -(2v/c^2) dv ⇒ v dv = -(c^2/2) du ∫ v (1 - v^2/c^2)^(-1/2) dv = ∫ u^(-1/2) (-(c^2/2) du) = -c^2 u^(1/2) = -c^2 sqrt(1 - v^2/c^2) ∫_{0}^{v} v (1 - v^2/c^2)^(-1/2) dv = [ -c^2 sqrt(1 - v^2/c^2) ]_{0→v} = -c^2 sqrt(1 - v^2/c^2) + c^2 = c^2 (1 - 1/γ) W = γ m v^2 - m c^2 (1 - 1/γ) = γ m v^2 - m c^2 + (m c^2)/γ 1/γ^2 = 1 - v^2/c^2 ⇒ v^2 + c^2/γ^2 = c^2 W = γ m (v^2 + c^2/γ^2) - m c^2 = γ m c^2 - m c^2 = (γ - 1) m c^2 K = W = (γ - 1) m c^2 E_total = K + m c^2 = γ m c^2 At v = 0 (γ = 1): E_rest = m c^2 일줄코드 승