Motion Under Constant Velocity
Recall that the position and the velocity of an object are related to each other by differentiation. If the position of an object is a function \(x(t)\), then the velocity of the object will be the derivative:
\[ v(t)=\dfrac{d}{dt}x(t).\]
Since the derivative of a function at a given point is the slope of the function at that point, the velocity of an object is the slope of the displacement function.
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Position-Time Graphs
Given a particular velocity function \(v(t)\), the position as a function of time can be constructed by integration. A position-time graph can then be constructed by plotting \(x(t)\) as a function in the usual way.
A particle moves with a velocity given by the function \(v(t) = -\sin (t)\). If the particle's initial position is \(x_0 = 2\), find and graph the position of the particle as a function of time.
Solution:
Integrating the given velocity function, one finds the solution:
\[x - x_0 = \int_0^t v(t') dt' \implies x(t) = 1 + \cos(t).\]
Graphing this function, one finds:
Graph of the function \(x(t) = 1+\cos (t)\).
Position-Time Graph for a Constant Velocity
When the velocity of a function is constant, the slope of the displacement function is also constant, i.e.
\[v(t)=\dfrac{d}{dt}x = \text{constant}.\]
Therefore, the graph of the velocity function will just be a straight line parallel to the \(x\)-axis at a constant value and the graph of the position function will be a line.
The motion of an object modeled by the linear position function \(x(t)=2t\) has the following graph:
Graph of the above given function
Graph the velocity of this object as a function of time.
Solution:
The derivative of the displacement function is the velocity function, thus: \[v(t)=\dfrac{d}{dt}x=\dfrac{d}{dt}(2t) = 2\]
Plotting this velocity yields a straight, horizontal line:
Velocity as a function of time; a straight horizontal line since the velocity is constant.
Position Vector for Constant Velocity
The velocity of a particle is constant if an object is moving equal distances at equal intervals of time and does not change its direction. Recall that velocity is defined as the derivative of the position function:
\[v(t)=\dfrac{dx}{dt},\]
i.e., it is the rate of change of the position over time.
From this equation, the change in position in a given time given a constant velocity \(v\) is simply:
\[\Delta x = vt.\]
If a particle starts at a position \(x_0\), the position as a function of time is therefore:
\[x(t) = x_0 + vt.\]
A particle begins at the position \(x=1\) and evolves according to the constant velocity \(v=-2\). Find the location of the particle as a function of time.
Solution:
The initial position is \(x_0 = 1\). Since the velocity is constant and equal to \(2\), the location of the particle as a function of time is:
\[x(t) = 1 + 2t.\]