Question

Alex, a rescue piot drops a survival kit while her plane is flying horizontally at an altitude of 1800.0 m with a forward velocity of 140.0 m/s. A group of lost Antarctica explorers is below on the ground. How far in front of the explorers should Alex drop the supplies?

Answer 1

The kit has position vector with components

$x=\left(140.0\,\dfrac{\mathrm m}{\mathrm s}\right)t$

$y=1800.0\,\mathrm m-\dfrac g2t^2$

It would take time $t$ for the kit to reach the ground from freefall:

$0=1800.0\,\mathrm m-\dfrac g2t^2\implies t=19.2\,\mathrm s$

It sounds like Alex's plane is 1800 m above the ground, at some (horizontal) distance away from the explorers below, and is heading in their direction. In order to have the kit drop down where the explorers are located, Alex should release the kit at a distance of

$x=\left(140.0\,\dfrac{\mathrm m}{\mathrm s}\right)(19.2\,\mathrm s)\approx2690\,\mathrm m$

away from the explorers.