Calculator of Alexandria

Newton's second law calculator (F = ma)

Calculate force, mass or acceleration with F = ma, in newtons, kilograms and m/s², with Newton's reasoning explained.

Solve for

Formula

F=m aF = m\,a
How it's derived

The formula

F=m aF = m\,a

FF is the net force in newtons (N), mm the mass in kilograms and aa the acceleration in m/s².

Where it comes from

Newton’s own statement, in the Principia (1687), was that the change in an object’s motion is proportional to the force applied. “Motion” meant what we now call momentum, p=mvp = mv. So force is the rate at which momentum changes:

F=dpdt=d(mv)dtF = \frac{dp}{dt} = \frac{d(mv)}{dt}

When the mass doesn’t change, it comes outside the derivative, and the rate of change of velocity is acceleration:

F=m dvdt=m aF = m\,\frac{dv}{dt} = m\,a

The newton is defined to make this work with no extra constant. One newton is the force that accelerates 1 kg at 1 m/s², so 1 N=1 kg⋅m/s21\ \text{N} = 1\ \text{kg·m/s}^2.

Worked example

A 1,200 kg car speeds up at 3 m/s²:

F=1200×3=3600 NF = 1200 \times 3 = 3600\ \text{N}

That’s the net force. The engine has to push harder than this to also overcome friction and air drag.

Watch out for

  • Net force. FF is the sum of all forces. A car cruising at a steady speed has zero acceleration, so zero net force, even with the engine running.
  • Mass, not weight. Weight is itself a force, mgmg. A 70 kg person weighs about 686 N on Earth.
  • Changing mass. Rockets burn fuel, so F=maF = ma alone doesn’t describe them; use the momentum form.

Sources

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