Lessons · projectile

Projectile motion: two 1D motions, superposed

Regime 1 — the algebra is the calculus, evaluated. Step the algebra, step the calculus, and watch the algebra formula fall out of the integral. SymPy proves the two registers agree.

Regime 1 · algebra is calculus, evaluated Math machine-derived & checked by SymPy 3 modeling assumptions (author-asserted)

A ball is launched from the ground at $v_0 = 20$ m/s, $\theta = 40°$ above the horizontal (no air resistance, $g = -10$ ). What is its range, how high does it go — and why is the path a parabola? Drag the launch angle and speed and watch the trajectory, range, and apex move together.

Range

R = v_0^2\sin 2\theta/|g|

39.39 m

Max height

H = (v_0\sin\theta)^2/2|g|

8.264 m

Time of flight

T = 2v_0\sin\theta/|g|

2.571 s

Time to apex

T/2

1.286 s

Split the launch into independent x and y motions

a_x = 0,\quad a_y = g \qquad v_{x0} = v_0\cos\theta,\quad v_{y0} = v_0\sin\theta

Gravity acts only vertically, so the horizontal and vertical motions are independent. Horizontally the acceleration is zero; vertically it is the constant $g$ — two constant-acceleration problems, side by side.

check $x''=0$ and $y''=g$ ; check the launch conditions; show the memorized $R$ and $H$ fall out of $x(t)$ and $y(t)$

✓ Horizontally there is no force, so $x'' = 0$ (constant velocity). [structural]
✓ Vertically the only acceleration is gravity, $y'' = g$ . [structural]
✓ It launches from the origin with the right velocity components: $x(0)=y(0)=0$ , $x'(0)=v_0\cos\theta$ , $y'(0)=v_0\sin\theta$ . [structural]
✓ The memorized range $R = v_0^2\sin 2\theta/|g|$ is exactly $x$ at landing — it falls out of the integral. [simplify]
✓ The memorized max height $H = (v_0\sin\theta)^2/2|g|$ is exactly $y$ at the apex. [structural]

Dimensional homogeneity: checked by SymPy (holds).

✗ Common misconception: “At the top of its arc, the projectile is momentarily at rest.”

Only the vertical velocity is zero at the apex; the horizontal velocity $v_0\cos\theta$ never changes — there is no horizontal force. At the top the ball is still moving horizontally at full speed. The horizontal and vertical motions are completely independent, which is exactly why the path is a parabola ( $y$ a quadratic in $t$ , $x$ linear in $t$ ).

Modeling assumptions — author-asserted, disclosed not discharged

No air resistance — the only force in flight is gravity (drag is the regime-2 sequel).
$g = -10$ m/s² (simplified from 9.81), up positive, so gravity carries its negative sign.
Point mass launched from and landing at the same height (flat ground).

The trajectory, fully annotated

A static rendering (Matplotlib) at the default launch — the path y vs x, with the apex and range marked. The interactive version with launch-angle and speed sliders is in the Graph tab above.

Formulas used

Hover a formula to preview its reference entry; click to open it in the reference (or the concept graph):