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
\( \dfrac{dx}{dt}\;dt=dx \)
\[
\begin{gather}
\int {{dx}}=v\int{{dt}}\\
x(t)+C_{1}=vt+C_{2}\\
x(t)=vt+C_{2}-C_{1}
\end{gather}
\]
C1 and C2 are integration constants that can be defined as a function of
a new constant C = C2 − C1.
\[
\begin{gather}
x(t)=vt+C \tag{I}
\end{gather}
\]
assuming that in the initial instant, t0, the particle is in the initial position
x0, we have the initial condition x(t0) = x0,
substituting in the expression (I)
\[
\begin{gather}
x(t_{0})=vt_{0}+C_{1}\\
x_{0}=vt_{0}+C_{1}\\
C_{1}=x_{0}-vt_{0} \tag{II}
\end{gather}
\]
substituting the expression (II) into expression (I)
\[
x(t)=vt+x_{0}-vt_{0}
\]
\[ \bbox[#FFCCCC,10px]
{x(t)=x_{0}+v\left(t-t_{0}\right)}
\]
what describes a particle in motion in a straight line.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .