Newton's Second Law

The first law tells us when motion changes (only when a net force acts). The second law tells us how much it changes. It is the quantitative heart of classical mechanics.

Concept

Newton's Second Law: The rate of change of momentum of a body is directly proportional to the net external force acting on it and takes place in the direction of the force.

$\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$

In SI units, the constant of proportionality is 1, defining the newton: $1 \, \text{N} = 1 \, \text{kg}\cdot\text{m/s}^2$.

When mass is constant (the usual case),

$\vec{F}_{\text{net}} = m\frac{d\vec{v}}{dt} = m\vec{a}$

Impulse: The product of force and the time over which it acts is called impulse:

$\vec{J} = \int_{t_1}^{t_2} \vec{F}\,dt$

The impulse-momentum theorem says impulse equals the change in momentum:

$\vec{J} = \Delta\vec{p} = \vec{p}_f - \vec{p}_i$

For a constant force, $\vec{J} = \vec{F}\,\Delta t$.

Derivation

Starting from the second law,

$\vec{F} = \frac{d\vec{p}}{dt}$

Integrate both sides from $t_1$ to $t_2$:

$\int_{t_1}^{t_2} \vec{F}\,dt = \int_{t_1}^{t_2} \frac{d\vec{p}}{dt}\,dt = \vec{p}(t_2) - \vec{p}(t_1)$

The left side is impulse $\vec{J}$ and the right side is $\Delta\vec{p}$:

$\vec{J} = \Delta\vec{p}$

This is why airbags, padded gloves, and bending knees on landing reduce injury — they extend $\Delta t$, lowering the force needed to produce the same momentum change.

Worked Example

A cricket ball of mass 0.15 kg moving horizontally at 30 m/s is caught by a fielder, who pulls his hands back during the catch, so that the ball comes to rest in 0.25 s. Find (a) the impulse on the ball and (b) the average force on the fielder's hands.

Solution:

(a) Impulse equals change in momentum:

$J = \Delta p = m(v_f - v_i) = 0.15(0 - 30) = -4.5 \, \text{kg·m/s}$

The magnitude is 4.5 N·s, opposite to the original motion.

(b) Average force:

$\bar{F} = \frac{\Delta p}{\Delta t} = \frac{-4.5}{0.25} = -18 \, \text{N}$

If the fielder had not pulled his hands back and the ball stopped in 0.05 s, the average force would be $-4.5/0.05 = -90 \, \text{N}$, five times higher — and painful.

Common Confusions

• "Force causes velocity." No — force causes acceleration, i.e. change in velocity. A constant force on a body gives constant acceleration, ever-changing velocity.
• Direction. $\vec{a}$ is along $\vec{F}_{\text{net}}$, not necessarily along $\vec{v}$. A ball thrown up has upward velocity but downward acceleration.
• Variable mass. For rockets and conveyor belts, mass changes, so the form $F = dp/dt$ must be used, not $F = ma$.
• Impulse is a vector. Don't add magnitudes; add components.

Key Takeaways

• $\vec{F} = d\vec{p}/dt$ is the most general form; $\vec{F} = m\vec{a}$ assumes constant mass.
• The newton is defined so the proportionality constant is 1.
• Impulse equals change in momentum: $\vec{J} = \Delta\vec{p}$.
• Increasing collision time reduces force — the principle behind airbags and cushioning.
• Acceleration is along net force, not along velocity.