Bug and Rivet Paradox Calculator
This calculator puts real numbers on the bug and rivet paradox, one of the sharpest thought experiments in special relativity. In the setup a rivet whose shaft is shorter than a hole is deep is fired at the hole at close to the speed of light, with a bug sitting safely at the bottom, or so it seems. Enter the speed as a fraction of the speed of light, the rest length of the rivet shaft and the depth of the hole, and the calculator returns the Lorentz factor, the contracted length of the shaft as seen in the hole's frame, the contracted depth of the hole as seen in the rivet's frame, and how far the shaft overshoots that contracted depth. The paradox is that the two frames seem to disagree: in the hole's frame the fast moving shaft is contracted and looks far too short to reach the bug, while in the rivet's frame the hole rushes towards it contracted and shallow, so the shaft clearly reaches the bug. Both cannot be the last word, and the resolution is that no material is perfectly rigid: when the rivet head stops, the tip keeps going until that information catches up at light speed, driving deep enough to squash the bug in every frame. The defaults fire a shaft 8 cm long into a 10 cm hole at 90 percent of light speed. Use it to see length contraction in action and to work through the paradox with numbers rather than hand waving.
In the hole's frame the contracted shaft looks too short to reach the bug, yet in the rivet's frame the shaft (8 cm) is far longer than the contracted hole, so it overshoots and reaches the bug. Because rigidity travels no faster than light, the tip drives home in every frame and the bug is crushed. Speeds must be below the speed of light.
How it works
The Lorentz factor gamma is one divided by the square root of one minus the speed squared, with speed measured as a fraction of light speed. Any length along the direction of motion is contracted to its rest value divided by gamma, but only in the frame where the object moves. In the hole's frame the shaft is moving, so its length is the rest shaft divided by gamma. In the rivet's frame the hole is moving, so the depth is the rest depth divided by gamma, while the shaft keeps its full rest length. The overshoot is that full shaft length minus the contracted hole depth, and a positive overshoot means the tip passes the bottom of the hole.
Worked example
Fire an 8 cm shaft into a 10 cm hole at 0.9 times light speed. Gamma is one divided by the square root of one minus 0.81, which is one divided by the square root of 0.19, about 2.294. In the hole's frame the shaft contracts to 8 divided by 2.294, about 3.49 cm, so it looks far too short. In the rivet's frame the hole contracts to 10 divided by 2.294, about 4.36 cm, while the shaft stays 8 cm, overshooting by 8 minus 4.36, about 3.64 cm. Both frames, worked through carefully, agree the bug is crushed.
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