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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