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Newton’s 2nd law (motion change)

• 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 2nd, so we can call it: Newton’s 2nd law. In his own words, the 2nd 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 2nd 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:

1. 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
2. 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
3. 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)