Time dilation is the gap that opens between a moving clock and a clock at rest. Set the speed with the β = v/c slider and the ship's own elapsed time with the t0 slider, then press Start: the two dials sweep at different rates and the panel reports the Lorentz factor, the dilated time and how far the clocks have drifted apart.
Start with the β = v/c slider, which runs from 0 to 0.999. It is the only control that changes the physics: every figure in the panel comes from gamma = 1 / sqrt(1 - v2/c2), so nudging β moves the Lorentz factor, the dilated time and the drift together. The speed readout restates the setting in kilometres per second — β = 0.600 is already 179,875 km/s. Press Start and the rest clock sweeps one full turn while the ship's clock manages only 1 / gamma of a turn; the shaded wedge between the hands is the gap you are being asked to find.
The second slider sets t0, the proper time aboard the ship, from 1 to 120 seconds. This one is a pure scale factor: leave β alone, drag t0 from 10 to 20, and the dilated time doubles while gamma does not move a digit, because the readout is solving t = t0 / sqrt(1 - v2/c2). So β sets the stretch and t0 sets how much there is to stretch. For your own numbers, including solving backwards for the speed, the time dilation calculator shows the steps.
The strip under the dials is the mechanism rather than the result. A pulse bounces between two mirrors and one bounce is one tick; because the ship also moves sideways, the pulse traces a zigzag whose diagonal beats the straight up-and-down path a resting clock uses. Light covers that longer path at the same speed c, so the tick takes longer. Watch the zigzag flatten as β rises: steep at 0.2, a wide shallow V at 0.9, barely bent at 0.999.
The misconception this lab corrects is that “fast” means “noticeable”. Drag β to 0.100 — about thirty thousand kilometres per second, far quicker than any spacecraft built — and gamma shifts by half a per cent, leaving the drift under a tenth of a second on a 10-second trip. Nothing dramatic happens until the last stretch of the slider, which is why the effect stayed hidden until particle accelerators arrived. For the wider theory, including length contraction, see the guide to special relativity.
It sets β = v/c, the ship's speed written as a fraction of the speed of light, from 0 up to 0.999. Nothing about the clocks is set directly; the slider only feeds the Lorentz factor, gamma = 1 / sqrt(1 - v^2/c^2), and every other readout follows from there. The panel also prints the same speed in kilometres per second so the number stays physical: β = 0.600 is about 179,875 km/s. Because gamma depends on β squared, the low end of the slider is almost inert and the last two per cent of the range does most of the work.
Because the speed enters the formula as v^2/c^2. At β = 0.1 that term is 0.01, so gamma = 1 / sqrt(0.99) = 1.005, a slowdown of about half a per cent. Even at a tenth of light speed, roughly 30,000 km/s, a 10-second trip aboard the ship registers as 10.05 seconds at rest. The curve only turns sharply upward past about β = 0.5, and the readouts show why: gamma is 1.15 at 0.5c, 1.67 at 0.8c and 22.37 at 0.999c.
The proper time t0 is the interval measured by a single clock that is present at both events, which here means the clock travelling with the ship. Enter the time the crew would read on their own watch. The simulator then returns the longer interval t recorded by the observer who stayed behind. Getting these the wrong way round is the most common mistake in these problems: the proper time is always the smaller of the two, so if your answer comes out shorter than the value you typed in, the inputs have been swapped.
No. This lab covers only the special-relativistic effect, where the difference between the clocks comes from relative motion in flat spacetime. Gravitational time dilation is a general-relativity result in which clocks deeper in a gravitational field run slow relative to clocks higher up, and it depends on the gravitational potential rather than on speed. Real systems can need both corrections at once, which is why GPS satellite clocks are adjusted for a velocity term and a gravitational term that push in opposite directions.
Because gamma has no value there. At β = 1 the quantity inside the square root becomes zero and 1 / sqrt(0) is undefined, so the dilated time would be infinite. That is not a limitation of the software but a statement about massive objects: accelerating one to the speed of light would take unbounded energy. The slider therefore stops at 0.999, where gamma is already 22.37 and one second aboard the ship stretches to more than 22 seconds at rest.