Ch 4.3 Projectile motion

Big ideas

• A projectile has $a_x = 0$ (constant horizontal velocity) and $a_y = -g$ (free fall vertically). Launch components: $v_{0x} = v_0\cos\theta$, $v_{0y} = v_0\sin\theta$.
• Time of flight comes from the vertical equation; horizontal range is then $x = v_{0x}t$. At the top $v_y = 0$ but $v_x$ is unchanged, so the speed at the top is $v_0\cos\theta$, not zero.
• Level ground shortcuts: $T = 2v_0\sin\theta/g$, $R = v_0^2\sin(2\theta)/g$ (max at $45^\circ$; complementary angles give equal ranges), $H = v_0^2\sin^2\theta/(2g)$.
• Horizontal launch from height $h$: $t = \sqrt{2h/g}$, $x = v_0 t$; impact speed $\sqrt{v_0^2 + 2gh}$ (energy works too).
• The trajectory is a parabola: $y = x\tan\theta - \dfrac{g x^2}{2v_0^2\cos^2\theta}$.

Key formulas

Worked example

Common mistakes

• Using the range formula when the landing height differs from the launch height.
• Saying the velocity is zero at the top (only v_y is).
• Forgetting that the horizontal speed never changes.