Relativistic Delivery Drone
A cargo drone travelling at 0.80c relative to a ground station carries a clock and a 2.0 m long antenna aligned along its direction of motion. Mission control on the ground measures the drone passing between two markers that are 120 m apart in the ground frame, and a ground clock records the transit. The drone's onboard clock records the same transit. A second, identical drone is stationary in the ground frame and its antenna is measured there. Engineers argue about which clock measures the 'proper' time and which length is the 'proper' length, and whether the two observers will agree on the drone's speed.
1. Identify the proper time and proper length in this scenario, and calculate the drone's speed as measured by each observer.
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Proper time: The drone's onboard clock is at rest relative to the two events (the antenna passing marker 1 and marker 2), so the onboard clock measures proper time. The ground clock measures a dilated interval.
Proper length: The antenna is at rest in the drone frame, so the drone measures its proper length (2.0 m). The ground observer measures a contracted length: L = L₀/γ where γ = 1/√(1−0.64) = 1.667, giving L = 2.0/1.667 = 1.20 m.
Speed: Relative speed is invariant. Both observers measure 0.80c. The ground observer uses ground distance/ground time; the drone observer uses contracted distance/contracted time. Since both length and time are scaled by the same factor, the ratio is unchanged.
2. The ground clock records a transit time of 5.0 × 10⁻⁷ s. Predict the time recorded by the drone's onboard clock and explain the apparent paradox that each observer sees the other's clock running slow.
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Ground frame: t = d/v = 120/(0.80 × 3.0 × 10⁸) = 5.0 × 10⁻⁷ s. This is the dilated time.
Drone frame: t₀ = t/γ = 5.0 × 10⁻⁷/1.667 = 3.0 × 10⁻⁷ s. Cross-check: in the drone frame the markers move past at 0.80c and are separated by 120/1.667 = 72 m, so t = 72/(2.4 × 10⁸) = 3.0 × 10⁻⁷ s. Consistent.
Reciprocity: Each observer's own clock is at rest with respect to themselves and therefore measures proper time between their own events. The 'other' clock is moving and is measured against two spatially separated events in the observer's frame, so simultaneity differs. There is no contradiction because the two observers are not comparing the same pair of events. The situation is symmetric only if neither accelerates; the drone must accelerate to return, breaking the symmetry.