Physics: Equation of Motions

Motions occur everywhere in our lives. Let say you are buying something in the supermarket, you are pushing a cart and you want to put the goods into the cart for the payment at the cashier.

How do you describe the motion?

Moving? Stopped?

Fast? Slow? It is too subjective.

Let me do some mathematical description for that.

Considering a 1-Dimensional movement, when a cart is moved from 1 point to another point for time t(in seconds). The displacement is from the original point changes as time goes towards t.

We may define the slope of the graph as the velocity.

We may also define the change of the slope as the acceleration.

In terms of the derivatives, we may write,

$a=\frac{dv}{dt} \quad or \quad v=\int_{}{} a \,dt$

$v=\frac{ds}{dt} \quad or \quad s=\int_{}{} v \,dt$

where $v$ is velocity, $a$ is acceleration, and $s$ is displacement

Now, let say if the acceleration is a constant, then:

(1) $\begin{equation*} \begin{split} v=\int_{0}^{t} a \,dt=at\\ v_1-v_0=v=at\\ v_1=v_0+at \end{split} \end{equation*}$

(2) $\begin{equation*} s=\int_{0}^{t} {v_1} \,dt=\int_{0}^{t} {v_0+at} \,dt=v_0{t}+\frac{1}{2} at^2 \end{equation*}$

where $v_0$ is initial velocity and $v_1$ is the final velocity

from equation 1 and 2, we may also derive that:

(3) $\begin{equation*} \begin{split} s=v_0{t}+\frac{1}{2} at^2\\ =(v_0+\frac{1}{2} at)(t)\\ =(v_0+(\frac{v_1-v_0}{2}))t\\ =\frac{1}{2} (v_0+v_1)t \end{split} \end{equation*}$

From (1) and (3)

(4) $\begin{equation*} \begin{split} s=\frac{1}{2} (v_0+v_1)t\\ =\frac{1}{2} (v_0+v_1)(\frac{v_1-v_0}{a})\\ =\frac{v_1^2-v_0^2}{2a} \\ v_1^2=v_0^2+2as \end{split} \end{equation*}$

The above equations are to describe the motion when the acceleration is constant.

But of course, it is impossible to happen in the real life.

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