What was astronaut's solution?

Question

Small space module has circular orbit around the Earth
with period T = 2 hours. Astronaut in open space gear
(total mass M = 120 kg) is on exactly the same orbit
d = 90 m behind the module. After completing his
calculations, astronaut tosses his calculator (m = 200 g)
with certain velocity, and after time T/2 = 1 hour reaches
his module.

What was astronaut's solution (speed and direction of the toss)?

If astronaut throws the calculator backward, his orbital speed will instantly increase, making his current position new perigee. One hour later he will find himself in apogee, on higher orbit, and further behind the module. After another one hour he will return to the original height, even further behind the module, because the new semi-axis is longer, and period is longer too. Soon poor fella will run outta oxigen.

Answer 1

15m/s in the opposite dir from his motion/orbit

treat module and astronaut as being a static frame of reference (not moving relative to one another). astronuat then has to catch up 90m in 1 hour, meaning he must go at 0.025m/s faster than module (from v=d/t). then we look at the conservation of momentum (moment before = moment after) as we are considering a static frame before we can assume total moment before = 0, therefore total moment (calculator + astronaut )= 0.
moment before = moment after
0 = (mass astro x v astro) + (mass calcultaor x v calculator)
- mass calc x veloc calc = mass astro x veloc astro
-0.2V = 120 x 0.025
-0.2V = 3
V = -15 m/s

assuming that the path of astronauts orbit doesn't chaneg when calc is thrown and speed alters - whiich it would do