Kinematics in Two Dimensions and Projectile Motion

2D Kinematics and Vectors

Fire a bullet horizontally and drop another simultaneously: they hit the ground together. Gravity is indifferent to horizontal motion, splitting every projectile problem into two independent stories sharing only the clock.

In 2D, position and velocity are vectors whose horizontal and vertical components act independently.

$$\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}, \quad \vec{v}(t) = \frac{d\vec{r}}{dt}$$

• The horizontal velocity $v_x = v_0\cos\theta$ is constant with zero acceleration.
• The vertical motion is free fall with $a_y = -g$ acting at all times.

Common pitfall: At the peak, the projectile is not force-free. Its vertical velocity hits zero, but gravity still acts at full strength, pulling it back down.

Projectile Formulas and Trajectory

For a projectile launched at speed $v_0$ and angle $\theta$, neglecting air resistance, the path is a parabola described by:

$$x(t) = v_0\cos\theta\,t, \quad y(t) = v_0\sin\theta\,t - \tfrac{1}{2}gt^2$$

• Maximum range occurs at $\theta = 45^\circ$ in a vacuum.
• Complementary angles ($\theta$ and $90^\circ - \theta$) yield the same range.