It is Christmas and, aboard the International Space Station, tipsy astronauts are celebrating by inflating and releasing some party balloons smuggled aboard by the newest arrival. Each balloon when empty has a mass of 2 g, can withstand a pressure of 0.05 bar, and can be inflated to a volume of 0.4 m. After a surreptitious vodka too many, the normally teetotal crew decide to don their spacesuits and release a balloon into space. Assume that the balloon maintains an approximately constant pressure as it deflates, and that the gaseous contents are accelerated by the pressure gradient in the neck of the balloon, which is 2 cm long. a) Making clear any assumptions, calculate the relative exit velocity of the air as it leaves the balloon. b) Hence determine the final speed achieved by the balloon. After remaining in orbit for some months, the balloon collides with a satellite following the opposite orbit. c) Calculate the kinetic energy of the balloon in the frame of the satellite. d) At what speed would a cricket ball of mass 160 g have the same kinetic energy? e) How, qualitatively, would your results differ if the pressure within the balloon fell as it deflated? At a pressure of 1 bar (10 Pa) and room temperature, air has a density of 1.23 kg m. The ISS orbits at 7.7 km s. It is Christmas and, aboard the International Space Station, tipsy astronauts are celebrating by inflating and releasing some party balloons smuggled aboard by the newest arrival. Each balloon when empty has a mass of 2 g, can withstand a pressure of 0.05 bar, and can be inflated to a volume of 0.4 m. After a surreptitious vodka too many, the normally teetotal crew decide to don their spacesuits and release a balloon into space. Assume that the balloon maintains an approximately constant pressure as it deflates, and that the gaseous contents are accelerated by the pressure gradient in the neck of the balloon, which is 2 cm long. a) Making clear any assumptions, calculate the relative exit velocity of the air as it leaves the balloon. b) Hence determine the final speed achieved by the balloon. After remaining in orbit for some months, the balloon collides with a satellite following the opposite orbit. c) Calculate the kinetic energy of the balloon in the frame of the satellite. d) At what speed would a cricket ball of mass 160 g have the same kinetic energy? e) How, qualitatively, would your results differ if the pressure within the balloon fell as it deflated? At a pressure of 1 bar (10 Pa) and room temperature, air has a density of 1.23 kg m. The ISS orbits at 7.7 km s
