Lessons · freefall

A ball thrown straight up

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 leaves your hand at 20 m/s straight up, from a height of 1.5 m. How high does it go, when does it land, and how fast is it moving when it hits the ground?

Time to the apex

2 s

Maximum height

21.5 m

Time of flight

4.074 s

Velocity at impact

-20.74 m/s

Start from the acceleration

a(t) = a

Constant — the only force acting is gravity. On the stacked graph this is a flat line.

simplify(algebra - calculus) == 0 for each identity below

✓ The algebra formula $v = v_0 + at$ is the integral $\int a\,dt$ with $v(0)=v_0$ . [structural]
✓ The algebra formula $x = x_0 + v_0 t + \tfrac12 a t^2$ is the integral $\int v\,dt$ with $x(0)=x_0$ . [structural]
✓ Velocity is zero at the apex time $t_{\text{apex}}$ — so the apex time is correct. [structural]
✓ Max height from the timeless equation equals the calculus trajectory at its apex. [structural]
✓ At the flight time the calculus trajectory returns to the ground — so the flight time is correct. [structural]
✓ Impact velocity from the timeless equation equals the calculus velocity at landing. [structural]

Dimensional homogeneity: checked by SymPy (holds).

✗ Common misconception: “At the top of the trajectory, the acceleration is zero (the ball 'stops').”

On the v-t graph the velocity passes through zero at the apex, but the line never bends: its slope is $a = -10$ m/s² the whole way. $v = 0$ for an instant; $a$ is never zero.

speeding up

\iff \operatorname{sign}(v\cdot a) > 0

; slowing down

\iff \operatorname{sign}(v\cdot a) < 0

rising

v + · a -

slowing down

apex

v 0 · a -

v = 0

, but

a \ne 0

falling

v - · a -

speeding up

Modeling assumptions — author-asserted, disclosed not discharged

No air resistance (the ball is in free fall under gravity alone).
g = -10 m/s^2, a clean-arithmetic simplification of -9.81 m/s^2.
The ball is a point mass; its size and spin are ignored.

The stacked graph, fully annotated

A static rendering (Matplotlib) at the default parameters — the interactive version 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):