5. A package is dropped at time `t = 0` from a helicopter that is descending steadily at a speed `v_i`. (a) What is the speed of the package in terms of `v_i, g`, and `t`? (b) What vertical distance `d` is it from the helicopter in terms of `g` and `t`? (c) What are the answers to parts (a) and (b) if the helicopter is rising steadily at the same speed?


Hint
`v_y = v_(yi) + a∆t`; `y_f = y_i + v_(yi)t + 1/2 a t^2`
Answer
(a)`color(red)(v_y = -v_i - g t)` (b)`color(red)(d = -1/2 g t^2)` (c) part (a) `color(red)(v_y = v_i - g t)` part (b) `color(red)(d = -1/2 g t^2)`
Show Steps
Set upward direction is positive. Known: dropped at time `t=0` and `a_y = -g` Given: `v_(yi) = -v_i` (a) `v_y = v_(yi) + a∆t` `=> color(red)(v_y = -v_i - g t)` (b) distance package moved ` = y_p` distance helicopter moved ` = y_h` distance between helicopter and package is: `d = y_p - y_h` `y_p = y_i + v_(yi)t + 1/2 a t^2` `y_h = y_i + v_(yi)t` where `y_i = 0` `d = -v_i t + 1/2(-g)t^2 - (-v_i t)` `color(red)(d = -1/2 g t^2)` (c) if `v_(yi) = +v_i` part (a) became `color(red)(v_y = v_i - g t)` part (b) still `color(red)(d = -1/2 g t^2)`