Projectile Motion

Equation of motion

\[ y = u_y t - \frac{1}{2} g t^2 \] \[ x = u_x t \] where \(u_x = u\cos\theta\) and \(u_y = u\sin\theta\).

Path (Trajectory)

Substituting \( t = x/u_x\) into \(y\), we see that \[ y = \tan\theta \cdot x - \frac{g}{2 u_x^2} \cdot x^2 \] No need to remember this equation.
But notice its form! \[ y = \boxed\cdots\, x - \boxed\cdots\, x^2 \]

Features of the form

1. Parabola ( \(x^2\) )

2. First \(\boxed\cdots \) describes the effect of projection angle
(or the slope of the projection line if no g)

3. Second \(\boxed\cdots \) describes the bending effect of vertical acceleration g

4. Minus (-) sign: dragged down by gravity

5. Dimension of the second \(\boxed\cdots\) must be 1/length, so that all terms have the same unit. In other words, [\(u_x^2/g\)] is some characteristic length.

6. The equation can also be written in a factorized form \[ y = \text{scaling factor} \times x(R-x) \] where \(R\) is the range, the max. \(x\).

Characteristic lengths and time

Half time of flight (time reaching the highest point): \[ \tau_* = \frac{u_y}{g} \] Max. height (max. y): \[ H = \frac{u_y^2}{2g} = \frac{1}{2} u_y \tau_* \] Range (max. x): \[ R = 2 u_x \tau_* = \frac{2u_x u_y}{g} \] In terms of \(R\), we can rewrite the trajectory as \[ y = \frac{\tan\theta}{R} \cdot x (R-x) \] Noting \( R\tan\theta = 4H\), we get \[ y = 4H \cdot \frac{x}{R} \left( 1 - \frac{x}{R} \right) \] which has the form \(y = A x' (1-x') \).

Natural scales of the system

Length: \( L = u^2/g \)

Time: \( T = u/g \)

Dimensionless form and dependency of \(\theta\)

write something