# 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 we sketch the graph velocity against time, the […]

# 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 we sketch the graph velocity against time, the […]

# 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 $a(t)=v'(t)=x^{\prime \prime}(t)$ $x(0)$, $v(0)$ and $a(0)$ give the position, velocity and acceleration […]

# 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 […]

# Applications of Maximum and Minimum

Example 1 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), […]