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# Motion on a Line

There is no point in using vectors for motion in a line. These are the equations of motion:

{\Large \begin{align} x &= a + bt +ct^2 \tag{position}\\ \dot x &= b + 2ct \tag{velocity} \\ \ddot x &= 2c \tag{acceleration} \end{align}}

Here, a is the initial position, b is the initial velocity, and 2c is the acceleration. These are not the equations most people learned in grade school. The a might be confused with acceleration, and grade school students cannot take derivatives, so they are provided with ready-made equations to be memorized.

Generally, the grade school formulas are obtained this way.

{\Large \begin{align} x &= a + bt +ct^2 \tag{position}\\ x - a &= bt + c^2t \tag{displacement} \\ d = s &= bt + c^2t \end{align}}

Exactly why some authors use s for displacement is not clear to me, but there is a tradition that uses s. This frees a to stand for acceleration, and for some reason we want acceleration to be coefficient free, so using the somewhat more mnemonic {$v_i$} we get:

{\Large \begin{align} d &= bt + c^2t \\ d &= v_i + \frac{1}{2}a^2t \tag{pupil's equation} \end{align}}

"Motion in a line"
Motion in a line

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Categories: Classical Mechanics

This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.