Newton's Second Law Calculator

Newton’s Second Law is one of the most fundamental equations in physics. It describes how forces change the motion of objects.

F = m x a

Force (F) in newtons equals mass (m) in kilograms times acceleration (a) in meters per second squared. Rearranged, it solves for any of the three variables:

a = F / m (acceleration from force and mass) m = F / a (mass from force and acceleration)

What makes this law profound is what it says about equilibrium. If F = 0, then a = 0 — constant velocity or rest. Every time you see an object moving at constant speed, all forces are balanced. Every time you see acceleration, there is a net unbalanced force.

Real-world examples:

• A 1,000 kg car accelerating at 3 m/s² requires a net force of 3,000 N from the engine (after subtracting friction and drag).
• Gravity accelerates all objects at 9.81 m/s² (on Earth’s surface). The force is F = mg — this is simply Newton’s Second Law applied to gravitational acceleration.
• A 70 kg skydiver in free fall feels 70 × 9.81 = 687 N (weight). Terminal velocity occurs when drag force = 687 N and net force = 0.

The law applies to the net force. If a 100 N applied force is opposed by 30 N of friction, the net force is 70 N and that is what drives the acceleration.

In imperial units, force is measured in pound-force (lbf), mass in slugs (1 slug = 14.59 kg), or you can use weight in pounds (lb) divided by 32.2 ft/s² for g.

The more general form: F = dp/dt. Newton actually wrote his second law as the rate of change of momentum, not as F = ma. The full statement is F = dp/dt, where p = mv is momentum. When mass is constant, the derivative reduces to F = m · dv/dt = ma. But when mass changes, you need the full form: F = m · dv/dt + v · dm/dt. The extra term v · dm/dt is what governs rocket propulsion.

A rocket loses mass continuously by expelling propellant. Setting up F = dp/dt for the rocket-plus-exhaust system and integrating gives the Tsiolkovsky rocket equation:

Δv = v_exhaust · ln(m_initial / m_final)

This is why rockets are mostly fuel by mass: doubling the velocity change requires multiplying the mass ratio by e (about 2.72×), not just 2×. SpaceX’s Falcon 9 is roughly 96% propellant at launch. F = ma alone can’t explain this; you need the variable-mass form.

Mass and weight are not the same. Mass (kg) measures the amount of matter and is constant everywhere. Weight is the force of gravity on that mass: W = mg. On the Moon (g ≈ 1.62 m/s²), your mass is unchanged but your weight is about one-sixth of what it is on Earth. A 70 kg person weighs 687 N on Earth but only 113 N on the Moon. Conflating mass with weight is the most common mistake in introductory mechanics.