- ‘A spaceship accelerates off into outer space.’ We can imagine the scenario. The rocket forces cause the spaceship to accelerate and speed up.
- ‘Suddenly the fuel is depleted, and the rocket engines turn off. The spacecraft stops accelerating. And just drifts.’ Clearly, acceleration only happens
*while the forces are pushing*. When the forces are gone, nothing accelerates anymore. Nothing speeds up, nothing slows down, nothing stops. Everything just drifts.

Clearly, forces cause acceleration. To be accurate: **the total force **(which we can write as $\sum \vec F$)** causes acceleration** ($\vec a$), since the forces add up. Also, *doubling* the total force *doubles* the acceleration, *tripling* it *triples* the acceleration etc. They appear to be **proportional**:^{1}

$$\sum \vec F\propto \vec a$$

How do we convert this proportionality into an equation? We need to figure out what the proportionality constant is. We might remember that the *mass* $m$ dampens acceleration, and it turns out that the mass $m$ indeed is our proportionality constant:^{2} ^{3}

$$\sum \vec F=m\vec a$$

Here, the $\sum \vec F$ term and the $m\vec a$ term are equal, so a larger mass causes a smaller acceleration.^{4}

Wait, how did we see that a larger mass causes a smaller acceleration from this simple formula? It is actually quite obvious: $m\vec a$ must remain equal to $\sum \vec F$. If $m$ is *larger*, $\vec a$ must be *smaller*. Otherwise the value of $m\vec a$ would be different and not equal to $\sum \vec F$ anymore. Push something heavy (with larger $m$) and it accelerates less than if you pushed something light (with smaller $m$). The mass “dampens” some of the force’s effort; it acts as a **resistance against acceleration**.

Let’s name this formula after the man who invented it: Sir Isaac Newton.^{5} He actually invented 3 laws, and this was his 2^{nd}, so we can call it: **Newton’s 2 ^{nd} law**. In his own words, the 2

^{nd}law is:

^{6}

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed

Isaac Newton^{[3]}

We must not forget that Newton’s 2^{nd} law is a *vector* relationship:

$$\sum \vec F=m\vec a\quad\Leftrightarrow\quad

\sum\begin{pmatrix}F_x\\F_y\end{pmatrix}=m \begin{pmatrix}a_x\\a_y\end{pmatrix} \quad\Leftrightarrow\quad

\begin{pmatrix}\sum F_x\\\sum F_y\end{pmatrix}= \begin{pmatrix}ma_x\\ma_y\end{pmatrix} $$

Meaning, we can split it along the different dimensions; it actually contains *two* equations (or three when in 3D):

$$\sum F_x=ma_x\\ \sum F_y=ma_y$$

So, we can choose to look only along the dimension we need; if we choose the dimension perpendicular to the stream in the example below (along the $x$-axis), we will thus use: $\sum F_x=ma_x$. And we will include only those forces and *parts of *forces that are along the x-axis.

References:

- ‘
**An experimental verification of Newton’s second law**’ (article),*Roberto Hessel, Saulo Ricardo Canola and others*, Revista Brasileira de Ensino de Física, vol. 35, issue 2, June 2013, www.researchgate.net/publication/275426838_An_experimental_verification_of_Newton’s_second_law, DOI 10.1590/S1806-11172013000200024 - ‘
**Historical development of Newton’s laws of motion and suggestions for teaching content**’ (article),*Wheijen Chang, Beverley Bell and others*, Asia-Pacific Forum on Science Learning and Teaching, vol. 15, issue 1, article 4, 2014, www.eduhk.hk/apfslt/download/v15_issue1_files/changwj.pdf - ‘
**Philosophiæ Naturalis Principia Mathematica**’ (book, English translation published 1728),*Isaac Newton*, 1st ed., vol. 1, 1687, en.wikisource.org/wiki/Page:Newton%27s_Principia_(1846).djvu/89 (accessed Sep. 27th, 2019)