Solved Problem on Kinematics
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Obtain the equation for displacement as a function of time with constant velocity from the expression of instantaneous velocity.
Solution
The instantaneous speed is given by
\[ \bbox[#99CCFF,10px]
{v=\frac{dx}{dt}}
\]
We integrate this expression in dt on both sides
\[
\int {{\frac{dx}{dt}\;dt}}=\int {{v\;dt}}
\]
as the speed v is constant, it is carried outside of the integral sign and
\( \dfrac{dx}{dt}\;dt=dx \).The limits of integration are x0, the initial position, and x(t), a position in an instant t for dx, and t0, initial time, and t, for dt
\[
\begin{gather}
\int_{x_{0}}^{x(t)}dx=v\int_{t_{0}}^{t}{{dt}}\\[5pt]
\left.x\;\right|_{\;x_{0}}^{\;x(t)}=v\;\left.t\;\right|_{\;t_{0}}^{\;t}\\[5pt]
x(t)-x_{0}=v\left(t-t_{0}\right)
\end{gather}
\]
\[ \bbox[#FFCCCC,10px]
{x(t)=x_{0}+v\left(t-t_{0}\right)}
\]
what describes a particle in motion in a straight line.
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