Time of Flight: The Symmetry of Up and Down

A projectile launched from ground level back to ground level has a beautiful symmetry: the time spent rising equals the time spent falling. The whole flight is two halves of the same motion, and the math gives us a clean formula that depends only on the vertical launch component and gravity.

The Physics

Vertical motion under gravity is independent of horizontal motion. The vertical velocity starts at $v_{0y} = v_0 \sin\theta$, decreases linearly under gravity, hits zero at the peak, then becomes negative (downward) at the same rate. By symmetry:

$t_{\text{down}} = t_{\text{up}} \qquad T_{\text{total}} = 2\,t_{\text{up}}$

Step-by-Step Solution

Given:

• Initial speed: $v_0 = 20\;\text{m/s}$
• Launch angle: $\theta = 30°$
• Gravity: $g = 9.81\;\text{m/s}^{2}$

Find: The total time the stone is in the air.

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Step 1 — Find the vertical component of the initial velocity.

$v_{0y} = v_0\sin\theta = 20\sin 30° = 20 \times 0.5 = 10.000\;\text{m/s}$

(The horizontal component $v_{0x} = 20\cos 30° \approx 17.32\;\text{m/s}$ does not affect time of flight at all — only $v_{0y}$ matters.)

Step 2 — Find the time to reach the peak.

The peak occurs when the vertical velocity reaches zero. Using $v_y(t) = v_{0y} - g\,t$:

$0 = v_{0y} - g\,t_{\text{up}}$

$t_{\text{up}} = \dfrac{v_{0y}}{g} = \dfrac{10.000}{9.81} \approx 1.0194\;\text{s}$

Step 3 — Use symmetry to find the total flight time.

Going up and coming down take the same amount of time (same speed at each height by energy conservation, just reversed direction):

$T = 2\,t_{\text{up}} = 2 \times 1.0194 \approx 2.039\;\text{s}$

Step 4 — (Bonus) Verify the height and impact velocity match symmetry.

The peak height is:

$H = v_{0y}\,t_{\text{up}} - \tfrac{1}{2}\,g\,t_{\text{up}}^{2} = (10)(1.0194) - \tfrac{1}{2}(9.81)(1.0194)^{2}$

$H = 10.194 - 5.097 \approx 5.097\;\text{m}$

The impact velocity has $v_x = 17.32\;\text{m/s}$ unchanged and $v_y = -10\;\text{m/s}$ (the negative of the launch value — that's the symmetry). Same speed up, same speed down.

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Answer: The stone is in the air for $T \approx 2.04\;\text{s}$, with $t_{\text{up}} = t_{\text{down}} \approx 1.02\;\text{s}$ for each half of the flight. The peak height is about 5.10 m.

A Surprising Consequence

Time of flight depends ONLY on $v_{0y}$ and $g$ — not on mass, not on horizontal velocity, not on shape. A bullet fired horizontally and a marble dropped from the same height hit the ground simultaneously. (Famously demonstrated by Galileo's thought experiments and modern bullet-drop classroom demos.)

Try It

• The two stopwatches show time-up (cyan) and time-down (pink) running in parallel — they finish at the exact same instant.
• Sweep the angle: at high angles the flight time stretches dramatically (since $\sin\theta$ grows toward 1 at $\theta = 90°$).
• Notice $T$ is independent of $v_0\cos\theta$ — only the vertical component $v_0\sin\theta$ enters the formula.