Which of Newton's laws is often written in modern form as F = dp/dt?

Newton’s second law is often written in modern form as F = dp/dt, meaning that net force equals the rate of change of momentum with respect to time. Here F is force and p is momentum, usually defined as mass multiplied by velocity.

For an object whose mass remains constant, p = mv, so differentiating gives F = m dv/dt = ma. This is the familiar classroom form F = ma. The momentum form is more general because it remains the natural statement when velocity changes and can also handle systems in which momentum is exchanged.

The law connects force with acceleration, not simply with motion itself. If the net force on an object is zero, its momentum remains constant: it may stay at rest or continue moving at constant velocity in a straight line. Forces must also be added as vectors, so opposing forces can cancel.

A common confusion is assigning F = dp/dt to the first law, because both laws concern unchanged motion when net force is zero. The first law describes the zero-net-force case; the second law gives the quantitative relationship between net force and changing momentum.