F=ma

More about F=ma

A futuristic humanoid figure races through a cosmic landscape, surrounded by planets and mathematical equations, symbolizing the fusion of science and human potential in the universe. As if an entrant in Cosmic Race of Science and Human Potential. Mathematically-speaking,
F = d(mv)/dt = m(dv/dt) + v(dm/dt)
When mass is constant, dm/dt = 0, and this reduces to the familiar F = ma since dv/dt is acceleration. But when mass varies (like in a rocket expelling fuel), the v(dm/dt) term becomes important.
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In relativistic notation, we express the force-momentum relationship using four-vectors and proper time. The relativistic force equation becomes:
F^μ = dp^μ/dτ
Where:

F^μ is the four-force
p^μ is the four-momentum (E/c, p_x, p_y, p_z)
τ is proper time
μ represents the four indices (0,1,2,3)

The four-momentum can be written as:
p^μ = γm₀(c, v_x, v_y, v_z)
Where:

γ = 1/√(1 - v²/c²) is the Lorentz factor
m₀ is the rest mass
c is the speed of light
v_x, v_y, v_z are the velocity components

This formulation maintains covariance under Lorentz transformations and reduces to F = dp/dt in the low-velocity limit where γ approaches 1.