# Differentiation and Displacement, Velocity and Acceleration

Distance Distance is the magnitude of the total movement from the start point or a fixed point. Displacement The displacement of a moving position relative to a fixed point. Displacement gives both the distance and direction that a particle is from a fixed point. For example, a particle moves $5$ units forwards from […]

# Kinematics using Integration

Distances from Velocity Graphs Suppose a car travels at a constant positive velocity $80 \text{ km h}^{-1}$ for $2$ hours. \begin{align} \displaystyle\text{distance travelled} &= \text{speed} \times \text{time} \\&= 80 \text{ km h}^{-1} \times 2 \text{ h} \\&= 160 \text{ km}\end{align}We sketch the graph velocity against time, the graph is a horizontal line, and […]

# Area Under a Curve using Integration

Distances from Velocity Graphs Suppose a car travels at a constant positive velocity $80 \text{ km h}^{-1}$ for $2$ hours. \begin{align} \displaystyle\text{distance travelled} &= \text{speed} \times \text{time} \\&= 80 \text{ km h}^{-1} \times 2 \text{ h} \\&= 160 \text{ km}\end{align}We sketch the graph velocity against time, the graph is a horizontal line, and […]

# 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: $$x(t) \xrightarrow{\text{differentiate}}v(t)=\dfrac{dx}{dt}\xrightarrow{\text{differentiate}} a(t)=\dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}$$ 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 […]

# Motion Kinematics

Displacement Suppose an object $P$ moves along a straight line so that its position $s$ from an origin $O$ is given as some function of time $t$. We write $x=x(t)$ where $t \ge 0$.$x(t)$ is a displacement function, and for any value of $t$ it gives the displacement from the origin.On the horizontal axis through […]

# Trigonometric Applications of Maximum and Minimum

Find the maximum value of $i=100\sin(50 \pi t +0.32)$, and the time when this maximum occurs. \( \begin{align} \displaystyle\dfrac{di}{dt} &= 0 \\100\cos(50 \pi t+ 0.32) \times 50 \pi &= 0 \\\cos(50 \pi t+ 0.32) &= 0 \\50 \pi t+ 0.32 &= \dfrac{\pi}{2}, \dfrac{3\pi}{2}, \cdots \\t &= \dfrac{1}{50 \pi}\Big(\dfrac{\pi}{2}-0.32\Big), \dfrac{1}{50 \pi}\Big(\dfrac{3\pi}{2}-0.32\Big) \cdots \\t &= 0.0080, 0.028, […]