$$ \begin{align} a &= \displaystyle \frac{d}{dx} \left( \frac{1}{2} v^2 \right) \\ v &= \frac{dx}{dt} \end{align} $$ Example A particle is moving so that the acceleration \( a = 32x^3 + 48 x^2 + 16x \). Initially \( x=1 \) and \( v=-8 \). Part 1 Show that \( \displaystyle a = \frac{d}{dx} \left( \frac{1}{2}v^2 \right) […]

# Motion

# Displacement, Velocity and Distance Travelled by Natural Logarithmic Equations

A particle is moving in a straight line, starting from the origin. At time t seconds the particle has a displacement of x metres from the origin and a velocity v. The displacement is described as a logarithmic expression in terms of time t. Find an expression for v, the initial velocity, when the particle comes to rest and the distance travelled by the particle in the first three seconds.

# 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), […]

# Collision of Projectile Motion

Collision of Projectile Motion is handled by setting the same vertical height and horizontal distance at a given time. Worked Examples of Collision of Projectile Motion Two Points \(A\) and \(B\) are \(d\) metres apart. A particle is projected from \(A\) towards \(B\) with initial velocity \(u\) m/s at angle \( \alpha \) to the […]