The Delivery Drone's Two-Stage Descent
A package-delivery drone descends vertically from rest at height 45 m. For the first 3.0 s its rotors produce a constant downward acceleration of 2.0 m/s². At t = 3.0 s the controller switches modes: the rotors now produce a constant upward acceleration of 4.0 m/s² (i.e., a net acceleration of 4.0 m/s² upward, gravity already included). The drone continues until it reaches the ground. A student claims the drone's speed monotonically increases the whole way down because 'it's falling.' Take downward as positive and ignore air resistance.
1. Determine the drone's velocity and height at t = 3.0 s, and decide whether the student's claim is correct for the first stage.
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Stage 1: a = +2.0 m/s², v₀ = 0, t = 3.0 s.
- v = at = (2.0)(3.0) = 6.0 m/s downward.
- Δy = ½at² = ½(2.0)(3.0²) = 9.0 m, so height = 45 − 9 = 36 m.
Speed rises from 0 to 6.0 m/s during stage 1, so the claim holds only for this stage. It is not a general truth - the claim must be tested stage by stage, not assumed from 'falling.'
2. Find the drone's velocity when it reaches the ground and identify whether its speed ever decreases during the descent.
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Stage 2: a = −4.0 m/s² (upward), v₀ = +6.0 m/s (down), displacement needed = +36 m.
- v² = v₀² + 2aΔy = 36 + 2(−4.0)(36) = 36 − 288 = −252.
- Negative v² is impossible → the drone never reaches the ground in stage 2; it decelerates to rest and reverses.
Time to stop: t = (0 − 6.0)/(−4.0) = 1.5 s. Distance up while stopping: ½(6.0)(1.5) = 4.5 m, so lowest point is 36 − 4.5 = 31.5 m, then it climbs. Speed decreases from 6.0 m/s to 0 during this interval, then increases upward - directly refuting the monotonic claim. The premise that it lands is false under the stated accelerations.
3. The engineer insists the drone 'must land' and blames the math. Reconcile the physical picture with the numbers.
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The math is consistent; the scenario is inconsistent with landing. With a net upward acceleration of 4.0 m/s² the drone behaves like a projectile thrown downward - it slows, stops at 31.5 m, and rises. For the drone to actually land, the stage-2 acceleration must be downward (or at least not strong enough to reverse motion before reaching y = 0). The judgment call: check whether the kinematic result is physically admissible (v² ≥ 0, consistent sign of velocity) before reporting a number. A negative v² signals a wrong assumption, not a computational slip.