Momentum. The momentum of a moving body is the product of its mass and its velocity. It tells us how much motion a body has and how hard it is to stop. Momentum is written as $p = mv$, where $m$ is the mass in kilograms and $v$ is the velocity in metres per second. Momentum is a vector quantity, pointing in the same direction as the velocity. Its SI unit is $\text{kg m/s}$ (kilogram metre per second). A heavy truck moving slowly and a light bullet moving fast can both carry large momentum, which is why both are hard to stop.
Newton's Second Law. The rate of change of momentum of a body is directly proportional to the applied unbalanced force and takes place in the direction of the force. In symbols, $F \propto \frac{\Delta p}{\Delta t}$. If a force $F$ acts for time $t$ on a body whose velocity changes from $u$ to $v$, then $\Delta p = mv - mu$ and $F = k\frac{\Delta p}{\Delta t} = \frac{m(v - u)}{t}$.
Deriving $F = ma$. Since acceleration $a = \frac{v - u}{t}$, we get $F = ma$ (taking the constant $k = 1$). This is the most useful form of the Second Law. It shows that:
- For a fixed mass, a larger force produces larger acceleration.
- For a fixed force, a larger mass produces smaller acceleration.
The newton defined. The constant $k = 1$ fixes the SI unit of force. One newton is the force that gives a mass of $1\ \text{kg}$ an acceleration of $1\ \text{m/s}^2$. So $1\ \text{N} = 1\ \text{kg m/s}^2$. Newton's First Law is actually a special case of the Second: when $F = 0$, $a = 0$, so the velocity stays constant.
Why force depends on time of contact. Because $F = \frac{\Delta p}{\Delta t}$, the same change in momentum needs a smaller force if the time is longer. This explains many real situations:
- A cricketer pulls his hands back while catching a ball, increasing $\Delta t$, so the force on his hands is reduced.
- Vehicles have crumple zones and we use airbags so that the stopping time increases and the force on passengers falls.
- A high jumper lands on a thick foam mattress so the longer stopping time lowers the impact force.
- Athletes are advised to bend their knees when landing to prolong contact and reduce force.