ENSC2004 - Lecture 11

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ENSC2004 - Lecture 10
Projectile Motion - Slides

Motion of a Projectile

• Projectile motion can be treated as two rectilinear motions
  • One in horizontal direction with zero acceleration
  • One in vertical direction experiencing constant acceleration
    $g=-9.81\text{m}{\text{s}}^{-1}$

$t=\frac{2v_0\sin (\theta +\alpha )}{g\cos \alpha }$

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Curvilinear Motion 2 - Slides

Normal and Tangential Components

• When a particle moves along a curved path, it is convenient to describe its motion using normal ($n$) and tangential ($t$) coordinates
  • The origin ($O$) is located on the particle, thus it alongside the coordinate system moves with the particle

• The centre of curvature ($O^{\prime }$) lies on the concave side of the curve
  • The $n$-axis has positive direction towards $O^{\prime }$
• The radius of curvature ($\rho$) is perpendicular distance from curve to $O^{\prime }$

The position of the particle at any instant is defined by $s$

• $s$ is the distance along the curve from a fixed reference point

Parameters in the n-t Coordinate System

• Velocity is tangent to path of motion: $v=\text{v}\cdot u_t$
  where $\text{v}=\dot{\text{s}}=ds/dt$
  • $\text{v}$ defines magnitude and $u_t$ defines direction
• Acceleration is the time rate of change of velocity: $a=\dot{\text{v}}{u}_t+\text{v}\dot{u_t}$
  where $\dot{\text{v}}$ represents change in magnitude of velocity and $\dot{u_t}$ represents rate of change in direction of $u_t$

Alternatively: $a=a_t{u}_t+a_n{u}_n=\sqrt{a_t^2+a_n^2}$

$a_t$ → tangential acceleration: $a_t=\frac{dv}{dt}$
i.e. $a_t$ is equal to the rate in change of speed

$a_n$ → normal (centripetal) acceleration: $a_n=\frac{v^2}{\rho }$
$\rho$ is the radius of curvature

Special Cases of Motion

1. Particle moves along a straight line
   $\rho \to \infty ⟹a_n=0⟹a=\dot{\text{v}}$
2. Particle moves with constant speed along curve
   $a_t=0⟹a=\frac{{\text{v}}^2}{\rho }$
3. $a_t$ is constant
   $s=s_o+v_{o}t+(1/2)(a_t{)}_c{t}^2$
   $v=v_o+(a_t{)}_{c}t$
   $v^2=v_o^2+2(a_t{)}_c(s-s_o)$
4. Particle moves along path $y=f(x)$
   $\rho =\frac{(1+{\frac{dy}{dx}}^2{)}^{\frac{3}{2}}}{|d^{2}y/d{x}^2|}$

Three-Dimensional Motion

• Binomial axis ($\text{b}$) is used when working in 3D space
  • It is perpendicular to the osculating plane
    Thus, $u_{\text{b}}$ is defined by cross product $u_{\text{b}}=u_t\times u_n$

No motion, thus no velocity or acceleration, in the binomial direction

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ENSC2004 - Lecture 12