The first law says zero net force means no change in velocity. The second law says what a net force does: it causes acceleration, and how much depends on the mass. Solving it is a one-step equation.
Key idea
Newton's second law: $$\Sigma F = m\,a$$ The net force (N) equals mass (kg) times acceleration (m/s²). So $a = \dfrac{\Sigma F}{m}$: more force, more acceleration; more mass, less.
Why it works
Push an empty shopping cart and a full one with the same force. The empty one speeds up quickly; the full one barely budges. Acceleration grows with the push and shrinks with the mass. Newton's law puts both in one equation, and the newton is defined to make it exact: $1\,\mathrm{N}$ gives $1\,\mathrm{kg}$ an acceleration of $1\,\mathrm{m/s^2}$.
Worked example: Find the acceleration
A 4 kg cart is pushed with a net force of 20 N. Find its acceleration.
- $a = \dfrac{\Sigma F}{m} = \dfrac{20\,\mathrm{N}}{4\,\mathrm{kg}}$.
- $a = 5\,\mathrm{m/s^2}$.
Answer: $5\,\mathrm{m/s^2}$
Worked example: With friction
A 50 kg crate is pushed with 180 N while friction pushes back with 80 N. Find its acceleration.
- Net force: $180 - 80 = 100$ N forward.
- $a = \dfrac{100\,\mathrm{N}}{50\,\mathrm{kg}} = 2\,\mathrm{m/s^2}$.
- Use the net force, not the push alone.
Answer: $2\,\mathrm{m/s^2}$
Common mistake
Thinking a steady force gives a steady speed. A steady net force gives a steady acceleration: the speed keeps growing by the same amount each second. You feel a steady speed only when the forces balance, for example when your push just matches friction. Check: if the net force isn't zero, the velocity must be changing.
Push a cart with a force probe in the Science Lab: The Second Law: F = ma.