$$ \text{Chapter 13: Projectiles} $$

Motion in vertical plane:

• Assumed no resistance, spin and gravity is constant
  • As it’s 2-dimension, acceleration is not constant
• $v=u+at\implies \begin{cases} v_x=u\cos \theta \\ v_y=u\sin\theta -gt \end{cases}$
  • Note that there is no force acting horizontally
• $v$ at a certain $t$ seconds $\displaystyle =\sqrt{v_x^2+v_y^2}$

<aside> 🍀 $u_x=u\cos\theta$

</aside>

<aside> 🍀 $u_y=u\sin\theta$

</aside>

• At the highest point, $v_y=0\to gt=u\sin\theta$

  $\implies t_{\text{at highest point}}=\frac{u\sin\theta}g\implies$

<aside> 🍀 total flight time $\displaystyle =\color{tomato}\frac{2u\sin\theta}{g}$

</aside>

• At the end of flight, $\displaystyle s_x=u\cos\theta.t+\frac 1 20t^2$

  $\implies s_x=u\cos\theta\frac{2u\sin\theta}{g}\implies$

<aside> 🍀 Horiztal Range$\displaystyle =\color{tomato}{\frac{u^2\sin2\theta}{g}}$

</aside>

• When particle is projected from raised platform, treat the vertical displacement $(s_y)$ below the given platform as negative

• With angle of depression, take downward direction as positive so displacement and acceleration downward are positive

Equation of trajectory:

• Relates between vertical and horizontal distance covered

<aside> 🍀 $\displaystyle y=x\tan\theta-\frac{gx^2}{u\cos\theta}\sec^2\theta$

</aside>

$\displaystyle y=0\implies x=\bigg[0,\frac{u^2\sin 2\theta}{g}\bigg]$

• When a projectile follows a parabolic path of $y=ax+bx^2$, equate coefficient with the equation of trajectory (make use of $sec^2\theta=1+\tan^2\theta$)

• Angle of direction of the particle changes throughout the motion as vertical velocity changes from greatest positive to 0 and to greatest negative while horizontal velocity stays the same

  • When particle has $\alpha\degree$ downward angle, $\displaystyle\color{tomato}\tan\alpha=\frac{v_y}{v_x}$

  $\implies (u\sin\alpha+gt)=\tan\alpha.(u\cos \alpha)$

• Separate horizontal and vertical components and use 5 Newton’s equations of motion

$$ \text{Chapter 14: Equilibrium of rigid body} $$

Moment of a force $(Nm)$:

• Calculate the distance in $\perp\triangle$ instead

<aside> 🍀 Moment of $F=(F.\cos\theta).d\\ =F.(\cos \theta$).d

</aside>

• The turning effect of a force on an object (spinning) of the point $O$
  • Only the perpendicular component has spinning effect
  • The forces that passes through $O$ has pulling/pushing effect
• Choose clockwise $(\circlearrowright)$ or anti-clockwise $(\circlearrowleft)$ as positive direction and a point which is the origin and Take moment about $O (\circlearrowleft/\circlearrowright)$

• Center of mass of a rod (without any external force acting on it) is always at the center