At a glance
A passenger on a carriage rolling at constant speed throws a ball straight up and catches it. The sim shows that one throw as seen from inside the carriage and from the platform, the Galilean starting point that special relativity later revises.
What you'll learn
- How the ball's path depends on which frame describes it.
- How to convert a position and a velocity from one frame to the other.
- Which of the flight time, the peak height and the distance along the track depend on v, u and g.
- What it means for two observers in uniform relative motion to be equivalent.
The carriage rolls at v along the track. The ball leaves the passenger's hand at speed u, straight up from the carriage.
What's on screen
- Ball (rose) with a white arrow for its velocity in the frame shown.
- Path: cyan in Train frame, amber in Platform frame. The bright part is flown so far, the faint part is still to come.
- Launch (green dot), the peak (hollow circle on the path) and land (red dot, Train frame only, shown briefly at each catch).
- Carriage, observer and lamp posts: the scenery slides past in Train frame, the carriage rolls past it in Platform frame.
- Position & velocity (top right): x, y, vx and vy of the ball in the frame shown.
- Peak-height line (dashed violet): appears when Show peak height is on.
See itStart one throw in the Train frame at your current v and u. Watch for: the ball leaving the hand, rising, and landing back in the same hand while the lamp posts slide past.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Frame | Which observer describes the throw: one riding in the carriage (Train frame) or one standing on the platform (Platform frame). The throw itself is the same. | 2 buttons opens in Train | |
| Train speed v | How fast the carriage rolls relative to the platform. | 10–30 m/s opens at 15 | |
| Throw speed u | How fast the ball leaves the hand, straight up, relative to the carriage. | 5–20 m/s opens at 10 | |
| Show peak height | Draws a dashed line at the greatest height the ball reaches. It appears in Free Exploration. | on / off opens off | |
| Position & velocity | The ball's live x, y, vx and vy in the frame shown. | overlay | |
| Symbols | A key to what v, u, x, y and g stand for. | panel opens folded | |
| ▶ Play / ⏸ Pause | Runs or freezes the throw. While running, the throw repeats. | top bar | |
| Speed | Plays the motion faster or slower on screen. The path and the numbers stay the same. | 0.25×–4× opens at 1× | |
| ↻ Reset | Restarts the mode you are in from its beginning. | top bar |
Fixed: g = 9.8 m/s², no air resistance, and a straight, level track.
One throw, two paths
Axes: x runs forward along the track, in the direction the carriage moves, and y is up. The platform frame is at rest with the track. The train frame rides with the carriage, so it moves at v relative to the platform.
The Galilean transform. Subtract the carriage's roll and the parabola straightens back out.
- Train frame. The ball leaves the hand with no horizontal velocity relative to the carriage. With no air resistance nothing pushes it sideways, so xtrain stays 0 and it rises and falls in a straight line.
- Platform frame. The ball already moves forward at v when it leaves the hand, and keeps doing so. Its x grows as vt while y rises and falls exactly as before.
- Together these make a parabola, y = (u/v)x − g x²/(2v²). The ball comes down in the hand because the carriage has covered the same distance, x = vT.
One throw, both frames, with the sim's defaults v = 15 m/s and u = 10 m/s.
See itWatch one throw in the Train frame, then the same throw in the Platform frame. Watch for: x staying at 0.00 in the first run and growing to about vT in the second, while the highest y is the same.
What the frame changes and what it leaves alone
Velocities transform by a plain shift of v along the track, and nowhere else.
- Along the track, the frame matters. x and vx shift by the carriage's motion. For this throw vx,train = 0, so vx,plat = v for the whole flight.
- Across the track, it does not. y, vy = u − gt and the acceleration −g are the same in both frames, because the frames differ only along the track.
- Time is the same. The Galilean model takes tplat = ttrain, so both observers measure the same flight time T. Special relativity drops this assumption.
The same velocity components at three moments. Cyan is vy, amber is vx.
See itStart in the Train frame and switch to the Platform frame mid-rise. The sim then pauses. Watch for: x and vx jumping at the switch, while y and vy keep their values.
Flight time, height and range
The ball leaves at vertical speed u and loses g every second (vy = u − gt). It stops rising at t = u/g, halfway through the flight.
Two numbers that belong to the throw, and one that belongs to the platform's view of it.
v
u
These are the sim's own Train speed and Throw speed sliders.
- T and ymax use only u and g. Doubling u doubles T and multiplies ymax by four. Changing v changes neither.
- R = vT = 2uv/g is the only one carrying v. Doubling v doubles R, and doubling u also doubles R.
R exists only in the Platform frame. In the Train frame the ball lands where it started, at x = 0.
See itRun the throw in the Platform frame at v = 10 m/s and then at v = 30 m/s, with the peak-height line on. Watch for: the dashed line and the highest y staying put while the path stretches along the track.
No experiment in the carriage shows v
In the Train frame, v never appears: xtrain, y(t), vy, T and ymax are built from u and g alone. A throw at v = 10 m/s and one at v = 30 m/s look the same from inside.
Lecture 6 notation: a tilde marks the train frame, and V is the carriage speed v.
- Constant v keeps the laws the same. The velocity transform adds a constant, so the acceleration is the same in both frames. If force and mass are also the same (the assumption of Lecture 6), both observers use F = ma.
- The observers are equivalent, not identical. Their descriptions differ, a line against a parabola, but neither is the more correct one (Lecture 0).
- Only constant velocity. If the carriage sped up or braked, the ball would no longer land in the hand.
What the frame switch leaves alone (left) and what it rewrites (right).
See itRun the throw in the Train frame at v = 10 m/s and then at v = 30 m/s. Watch for: the scenery sliding past at different speeds while the ball's height and the readout values stay the same.
Where it matters and where it stops
- Everyday speeds. Adding velocities, w = w′ + v, works: for w′ = 8 m/s on a carriage at v = 12 m/s it differs from the relativistic rule by about one part in 10¹⁵.
- The principle survives. In Lecture 0's ship, a fly dropped in the cabin falls straight for Alice and Bob and along a parabola for Galileo on the beach. Special relativity keeps this equivalence of observers.
- Light breaks the rule. A light pulse leaving the carriage at c would reach the platform at c + v by w = w′ + v. Michelson and Morley found no such difference (Lecture 7): light moves at c in every frame, so velocity addition and absolute time must change.
The ball's velocities add. The pulse's do not. Illustrative numbers: w′ = 8 m/s on a carriage at v = 12 m/s.
Nothing in this sim is relativistic: there is no Lorentz factor, length contraction or time dilation.
Go deeper: relative time at high speed
- An atom crossing 1 light-second (about 300,000 km) at 0.1c takes 10 s by the ground clock and about 9.95 s by a clock moving with it.
- An electron at 0.9c takes 1.11 s by the ground clock and about 0.48 s by its own.
Summary
In one line. Between frames moving at constant velocity, a Galilean description shifts x by vt and vx by v, and leaves y, vy, the acceleration and time alone.
Key points
- One throw is a straight line in the Train frame and a parabola in the Platform frame.
- x and vx shift by vt and v. y, vy and t are the same in both frames.
- T = 2u/g and ymax = u²/(2g) belong to the throw. R = vT belongs to the platform's view.
- At constant v nothing inside the carriage reveals v. Velocity addition holds at everyday speeds and fails for light.
Common mix-ups
- "The ball goes straight up in both frames." Only in the Train frame. From the platform it moves forward at v throughout.
- "The carriage rolls out from under the ball." The ball keeps the carriage's forward speed, so it stays above the hand.
- "The platform's clock reads longer because the path is longer." In the Galilean model both clocks read T = 2u/g.
Things to try on your own
- Pause a throw in the Train frame and note the four readouts. Switch to the Platform frame and list which changed.
- Set u = 20 m/s and v = 10 m/s. Work out T, ymax and R by hand (4.08 s, 20.4 m, 40.8 m) and check them against section 05.
- Run v = 10 m/s and v = 30 m/s in the Train frame and list any readout value that differs.