When we apply Newton’s Second Law in one dimension, we must realize that it is a vector equation. This means that the direction matters. Remember from kinematics in one dimension, the “forward” direction is positive, and the “backward” direction is negative.

Example of Newton’s Second Law in One Dimension

Let’s look at a situation in which two forces are acting on an object. We will choose the positive direction to follow the usual mathematical convention of pointing to the right. The first force \(\color{black}{\vec{F}_1}\) points left (in the negative direction). The second force \(\color{black}{\vec{F}_2}\) is a little smaller and points to the right (the positive direction).

Two forces applied to an objectWe should draw a free body diagram of these two forces, which will look something like this…

Once this is done, we write down Newton’s Second Law in vector form as our starting equation.

\(\color{black}{m \vec{a} = \Sigma \vec{F}}\)


Notice that I wrote it down with the \(\color{black}{m \vec{a}}\) first. This helps keep my equal signs lined up on the page, since we won’t do much with this half of the equation.

Now we write down the forces that make up the left hand side of the equation.

\(\color{black}{m \vec{a} = \vec{F}_1 + \vec{F}_2}\)

Notice that I have left the equation as an addition. We still have a vector equation here. Next, we’ll put the positives and negatives, creating an algebra equation.

\(\color{black}{m a = {}^-F_1 + {}^+F_2}\)

I wrote the direction signs a little higher than normal just so we don’t lose them.