Kinematics

Kinematics describes motion without asking why it happens.

1. Position, velocity and acceleration as vectors (Def. 1.1).
2. Constant acceleration (Prop. 1.2) and projectiles (Example 1.3, Prop. 1.4).
3. Polar coordinates (Prop. 1.5), the natural language for circular and orbital motion.

A particle's position is $\mathbf{r}(t)$; its velocity $\mathbf{v} = \dot{\mathbf{r}}$ (tangent to the path; its length is the speed) and acceleration $\mathbf{a} = \dot{\mathbf{v}} = \ddot{\mathbf{r}}$. In Cartesian components each is differentiated (or integrated) componentwise. Integrating back needs initial conditions: $\mathbf{v}(t) = \mathbf{v}_0 + \int_0^t\mathbf{a}$, $\mathbf{r}(t) = \mathbf{r}_0 + \int_0^t\mathbf{v}$.

If $\mathbf{a}$ is constant: $\mathbf{v} = \mathbf{u} + \mathbf{a}t$ and $\mathbf{r} = \mathbf{r}_0 + \mathbf{u}t + \frac12\mathbf{a}t^2$. In one dimension also $v^2 = u^2 + 2as$ and $s = \frac12(u + v)t$ (the "suvat" equations).

Ignoring air resistance, a projectile launched with speed $u$ at angle $\alpha$ has $\mathbf{a} = (0, -g)$, so

Time of flight $\frac{2u\sin\alpha}g$, maximum height $\frac{u^2\sin^2\alpha}{2g}$, range $\frac{u^2\sin2\alpha}g$ (on level ground), maximised at $\alpha = 45°$. Eliminating $t$, the path is a parabola.

To hit the point $(X, Y)$ with launch speed $u$, the angle satisfies (with $T = \tan\alpha$)

a quadratic in $T$: two angles (high and low) if the discriminant is positive, one if zero, none (out of reach) if negative.

With $\hat{\mathbf{r}} = (\cos\theta, \sin\theta)$, $\hat{\boldsymbol\theta} = (-\sin\theta, \cos\theta)$ and $\mathbf{r} = r\hat{\mathbf{r}}$:

The transverse acceleration equals $\frac1r\frac{\mathrm{d}}{\mathrm{d}t}(r^2\dot\theta)$.

• Choose axes and a positive direction; keep signs consistent ($g$ acts downwards).
• Constant acceleration only for "suvat"; otherwise integrate.
• Projectiles: treat $x$ and $y$ separately; time links them.
• Check units and limiting cases ($\alpha = 0$, $\alpha = 90°$).