Independence of motions, put to the test.
One ball drops. Another is launched horizontally from the same height at the same instant. It seems obvious that the one thrown “further” should take longer — and yet they hit the ground together, every time. Here you can change the launch speed and the height and check it yourself.
A ball dropped and a ball launched horizontally from the same height hit the ground at the same instant.
The principle
The interactive figure
Use the arrows on the figure (or swipe) to follow the story step by step, then drag the v₀ and h sliders — in the last step a time slider appears as well, so you can scrub through the fall by hand. On a keyboard, ← and → change step.
Hover over (or tap) a ball, the guide line or the ground to find out what it is.
Two balls, same height: which hits the ground first?
Ball A is dropped. Ball B is launched horizontally, with a speed v₀, from the same height at the same instant. Intuitively it seems that B, since it also has to “go forward”, should take longer. Try placing your bet before you continue.
fall time = 1.75 sdistance of B = 12.25 mheight fallen = 0.00 m
With h = 15 m the fall time is 1.75 s for BOTH balls, whatever v₀ = 7 m/s is.
The reasoning, step by step
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Step 1 · The bet
Two balls, same height: which hits the ground first?
Ball A is dropped. Ball B is launched horizontally, with a speed v₀, from the same height at the same instant. Intuitively it seems that B, since it also has to “go forward”, should take longer. Try placing your bet before you continue.
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Step 2 · The launch
A falls, B starts with a horizontal push
A has zero initial speed and moves only vertically. B starts with the same zero vertical speed, but also with a horizontal speed v₀ that never changes during the fall — no force changes it, because gravity is vertical.
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Step 3 · Same vertical law
Vertically, A and B obey the same law: y = ½gt²
Gravity pulls both balls down in exactly the same way, regardless of how fast they move horizontally. The height fallen after a time t is ½gt² for A and, separately, ½gt² for B too: the same formula, the same number, at every instant.
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Step 4 · Horizontal motion
x = v₀t describes ONLY where B goes, never how far it falls
The horizontal distance covered by B grows uniformly, x = v₀t — but this formula only concerns the horizontal position. It does not enter anywhere in the calculation of how far B has fallen: the two motions, vertical and horizontal, really are independent of each other.
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Step 5 · The landing
They hit the ground together
Precisely because the vertical fall does not “know” anything about the horizontal motion, the fall time is the same for A and for B: it depends only on the starting height, never on v₀. B lands further away than A, but at exactly the same instant.
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Step 6 · Try it
Change v₀ and the height, and watch the landing instant
Raise v₀ as much as you like: ball B will land ever further away, but never later or sooner than A. Only changing the height h changes the fall time — and it changes it for BOTH balls in the same way.
What each element of the drawing is
In the figure these explanations appear when you hover over the element. Here they are all written out.
- Ball A — free fall
- Zero initial speed, moves only vertically: y(t) = ½gt².
- Ball B — horizontal launch
- Same vertical law as A, y(t) = ½gt², plus a uniform horizontal motion x(t) = v₀t that neither slows it down nor speeds it up as it falls.
- The horizontal guide line
- It joins the two balls at the same height, at every instant of the drawing. If the horizontal motion affected the vertical one, this line would not stay horizontal — but it always does.
- The ground
- Where both balls land — at the same instant, whatever v₀ is.
The questions we always get.
Why does v₀ not affect the fall time?
Because gravity acts only vertically, and the horizontal acceleration is zero for both balls. The two components of the motion — horizontal and vertical — are calculated with independent formulas: changing v₀ only changes x(t), never y(t).
Is it true in real life too, with air resistance?
In a perfect vacuum, yes, exactly. With real air the drag depends on the ball’s total speed, which is higher for B — so in practice B slows down slightly more and lands a fraction of a second after A. The effect is small for compact objects and moderate speeds, which is why the experiment “works” outside the laboratory too.
What changes if I throw B upwards instead of horizontally?
Everything changes: an upward launch has an initial vertical component that is not zero, so the law of vertical motion is no longer ½gt² but also includes that initial speed. The independence of motions always holds between the two components, but here both components would be vertical.
Keep going from here
Projectile motion never clicks?
Splitting a motion into two independent components is the idea that unlocks all two-dimensional kinematics — from horizontal launches to parabolic motion. It can be recovered in a few hours, with the right practice.