# How to Express the Velocity and the Acceleration as Functions of Displacement and Time

If a particle the displacement gives $P$ moves in a straight line and its position 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 $a(t)=v'(t)=x^{\prime \prime}(t)$
- $x(0)$, $v(0)$ 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 measures how fast something is travelling, regardless of the direction of travel. Speed is a scalar quantity that 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.

\( \begin{align} \displaystyle

v(t) &= x'(t) \\

&= 6t^2 + 6t

\end{align} \)

(b) Find the expression for the particle’s acceleration.

\( \begin{align} \displaystyle

a(t) &= 12t + 6

\end{align} \)

(c) Describe the motion of its initial conditions.

\( \begin{align} \displaystyle

x(0) &= 2 \times 0^3 + 3 \times 0^2 -6 \\

&= -6 \\

v(2) &= 6 \times 0^2 + 6 \times 0 \\

&= 0 \\

a(0) &= 12 \times 0 + 6 \\

&= 6

\end{align} \)

The particle is 6 metres to the left of the origin, at rest initially, then moving to the right.

### Example 2

Represent the acceleration expression in terms of displacement and velocity.

\( \begin{align} \displaystyle a &= \frac{dx}{dt} \times \frac{dv}{dx} \\ &= v \times \frac{dv}{dx} &\require{ASMsymbols} \color{green}{\text{as } v = \frac{dx}{dt}} \\ &= \frac{d}{dv} \frac{v^2}{2} \times \frac{dv}{dx} \\ &= \frac{d}{dx} \frac{v^2}{2} \end{align} \)

### Example 3

Represent the acceleration expression in terms of displacement and time.

\( \begin{align} \displaystyle a &= \frac{dx}{dt} \times \frac{dv}{dx} \\ &= \frac{dx}{dt} \times \frac{d}{dx}v \\ &= \frac{dx}{dt} \times \frac{d}{dx}\frac{dx}{dt} & \require{ASMsymbols} \color{green}{\text{as } v = \frac{dx}{dt}} \\ &= \frac{d}{dt}\frac{dx}{dt} \\ &= \frac{d^2 x}{(dt)^2} \\ &= \frac{d^2 x}{dt^2} \end{align} \)

**✓ **Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions!

**✓ **Discover more enlightening videos by visiting our YouTube channel!

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

## Responses