Independence of Horizontal and Vertical Motion: Why Two Balls Land Together (IB Physics)
Drop one ball and throw another sideways from the same height at the same moment. Which lands first? Almost everyone guesses the dropped one, and almost everyone is wrong. The two balls land together, and the reason is the single most useful idea in projectile motion: horizontal and vertical motion are independent.
The experiment that surprises everyone
Ball A is released from rest. Ball B leaves the same height at the same instant, but it is thrown horizontally. B travels a long way sideways, so it feels as though it must be in the air longer. It is not. Film them side by side and the two balls are at the same height in every frame, right down to the ground.
Why it works: two separate motions
Gravity pulls straight down. It does nothing sideways. So the motion splits cleanly into two parts that never interfere with each other.
- •Horizontal motion: no force, so acceleration a = 0 and the velocity stays constant. Distance is simply s = ut.
- •Vertical motion: gravity only, so a = g downward. A ball released horizontally has zero initial vertical velocity, the same as a ball that is just dropped.
- •The only thing the two motions share is time. The same clock runs for both axes. That is how you connect them: find the time from one axis, then use it in the other.
The equations you need
- •Horizontal: s = ut (constant velocity).
- •Vertical, starting from rest vertically: s = ½gt², v = gt, v² = 2gs.
- •Combine the two parts of a vector with Pythagoras, because horizontal and vertical are at right angles: resultant = √(x² + y²).
Worked example (take g = 10 m/s²)
Two small balls start together from a height of 50 m. A is dropped. B is projected horizontally at 5 m/s. Air resistance is negligible.
Q1, the time for A to land: use s = ut + ½at² with u = 0. 50 = ½ × 10 × t², so t² = 10 and t = √10 = 3.16 s.
Q2, how far across B lands: B takes the same 3.16 s, and horizontal motion is uniform, so R = 5 × 3.16 = 15.8 m.
Q3, the displacement of B from start to landing: the horizontal part is 15.8 m and the vertical part is 50 m, at right angles. d = √(15.8² + 50²) = √2750 = 52.4 m, at about 72.5° below the horizontal.
Common mistakes
- •Thinking the thrown ball falls more slowly because it is moving sideways. Sideways speed changes where it lands, not when.
- •Using the horizontal speed in the vertical equation, or the other way round.
- •Giving the length of the curved path when the question asks for displacement, which is the straight line from start to landing.
- •Writing the horizontal acceleration as g. It is zero when air resistance is ignored.
Try it: drop, throw and play
Our free DP Physics lesson on independence of motion has an animation of this exact problem where you can change the speed and the height, a worked solution for all three questions, and a "supply drop" game where you decide when to release a crate so that it lands on a target. The key to winning is the same two steps as above: find the time to fall, then the distance travelled sideways in that time.
Frequently asked questions
Yes, if air resistance is negligible and they start from the same height. The time to fall depends only on the height and g, because gravity acts only vertically and both balls start with zero vertical velocity.
It means the horizontal motion (constant velocity) and the vertical motion (constant acceleration g) do not affect each other. You can solve them separately and link them only through the time.
Use s = ut + ½at² on the vertical axis with u = 0, which gives t = √(2h / g). For 50 m and g = 10 m/s² that is 3.16 s.
In real life it does slightly, which is why a feather and a stone differ. In IB Physics projectile problems air resistance is stated to be negligible, so the result holds exactly.