Velocity and Acceleration

If a particle $P$ moves in a straight line and its position is given by the displacement function $x(t)$, then:
  • the velocity of $P$ at time $t$ is given by $v(t) = x'(t)$
  • the acceleration of $P$ at time $t$ is given by $a(t)=v'(t)=x''(t)$
  • $x(0)$, $v(0)$ and $a(0)$ give the position, velocity and acceleration of the particle at time $t=0$, and these are called the initial conditions.
$$x(t) \xrightarrow{\text{differentiate}}v(t)=\dfrac{dx}{dt}\xrightarrow{\text{differentiate}} a(t)=\dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}$$

Sign Interpretation

Suppose a particle $P$ moves in a straight line with displacement function $s(t)$ relative to an origin $O$. Its velocity function is $v(t)$ and its acceleration function is $a(t)$.
The sign diagram is being used to interpret:
  • where the particle is located relative to the origin
  • the direction of motion and where a change of direction occurs
  • when the particle's velocity is increasing or decreasing

Sign of Displacement $x(t)$

\( \begin{array}{|c|c|} \hline x(t)=0 & \text{the particle is at the origin} \\ \hline x(t) \gt 0 & \text{the particle is located at the right of the origin} \\ \hline x(t) \lt 0 & \text{the particle is located at the left of the origin} \\ \hline \end{array} \)

Sign of Velocity $v(t)$

\( \begin{array}{|c|c|} \hline v(t)=0 & \text{the particle is at rest} \\ \hline v(t) \gt 0 & \text{the particle is moving to the right} \\ \hline v(t) \lt 0 & \text{the particle is moving to the left} \\ \hline \end{array} \)

Sign of Acceleration $a(t)$

\( \begin{array}{|c|c|} \hline a(t)=0 & \text{velocity is increasing} \\ \hline a(t) \gt 0 & \text{velocity is decreasing} \\ \hline a(t) \lt 0 & \text{velocity is constant} \\ \hline \end{array} \)

Speed

Velocities have magnitude and direction. In contrast, speed simply measures how fast something is travelling, regardless of the direction of travel. Speed is a scalar quantity which has size but no sign. Speed is always positive, but cannot be negative.
\( \begin{array}{|c|c|c|c|} \hline v(t) & a(t) & \text{movement} & \text{speed} \\ \hline + & + & v=1, 2, 3, \cdots & s=1, 2, 3, \cdots \text{ increasing} \\ \hline - & - & v=-1, -2, -3, \cdots & s=1, 2, 3, \cdots \text{ increasing} \\ \hline + & - & v=4, 3, 2, \cdots & s=4, 3, 2, \cdots \text{ decreasing} \\ \hline - & + & v=-4, -3, -2, \cdots & s=4, 3, 2, \cdots \text{ decreasing} \\ \hline \end{array} \)

Example 1

A particle moves in a straight line with position relative to the origin given by $x(t)=2t^3 + 3t^2 -6$ metres, where $t$ is the time in seconds.
(a) Find the expression for the particle's velocity. (b) Find the expression for the particle's acceleration. (c) Describe the motion of its initial conditions.