Uniform Linear Motion Equations Deduction

September 27, 2024

This is a proof of the equations of uniform linear motion. We will integrate the acceleration to obtain the velocity and the position of a particle in uniform linear motion.

To prove the equations of uniform linear motion, we will integrate the acceleration to obtain the velocity and the position of a particle in uniform linear motion.

First of all, let’s consider a particle moving in a straight line with a constant acceleration aa. We will denote the initial velocity as v0v_0 and the initial position as x0x_0. We know that the acceleration represents the rate of change of velocity with respect to time, and the velocity represents the rate of change of position with respect to time. So we have the following relationships:

a=dvdtv=dxdt\begin{align*} a &= \frac{dv}{dt} \\ v &= \frac{dx}{dt} \end{align*}

Integrating the acceleration with respect to time, we obtain the velocity as a function of time:

∫a dt=∫dvdt dtat+C1=v\begin{align*} \int a \, dt &= \int \frac{dv}{dt} \, dt \\ a t + C_1 &= v \end{align*}

where C1C_1 is the constant of integration. We can determine the value of C1C_1 by using the initial condition v(t=0)=v0v(t=0) = v_0:

a⋅0+C1=v0C1=v0\begin{align*} a \cdot 0 + C_1 &= v_0 \\ C_1 &= v_0 \end{align*}

Therefore, the velocity of the particle as a function of time is given by:

v=v0+atv = v_0 + a t

Integrating the velocity with respect to time, we obtain the position as a function of time:

∫v dt=∫dxdt dtvt+C2=x\begin{align*} \int v \, dt &= \int \frac{dx}{dt} \, dt \\ v t + C_2 &= x \end{align*}

where C2C_2 is the constant of integration. We can determine the value of C2C_2 by using the initial condition x(t=0)=x0x(t=0) = x_0:

v⋅0+C2=x0C2=x0\begin{align*} v \cdot 0 + C_2 &= x_0 \\ C_2 &= x_0 \end{align*}

Therefore, the position of the particle as a function of time is given by:

x=x0+v0t+12at2x = x_0 + v_0 t + \frac{1}{2} a t^2