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Professional Application
During a visit to the International Space Station, an astronaut was positioned motionless in the center of the station, out of reach of any solid object on which he could exert a force. Suggest a method by which he could move himself away from this position, and explain the physics involved.
Professional Application
It is possible for the velocity of a rocket to be greater than the exhaust velocity of the gases it ejects. When that is the case, the gas velocity and gas momentum are in the same direction as that of the rocket. How is the rocket still able to obtain thrust by ejecting the gases?
Professional Application
Antiballistic missiles (ABMs) are designed to have very large accelerations so that they may intercept fast-moving incoming missiles in the short time available. What is the takeoff acceleration of a 10,000-kg ABM that expels 196 kg of gas per second at an exhaust velocity of $2\text{.}\text{50}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s?}$
$\text{39}\text{.}2\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$
Professional Application
What is the acceleration of a 5000-kg rocket taking off from the Moon, where the acceleration due to gravity is only $1\text{.}6\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ , if the rocket expels 8.00 kg of gas per second at an exhaust velocity of $2\text{.}\text{20}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s?}$
Professional Application
Calculate the increase in velocity of a 4000-kg space probe that expels 3500 kg of its mass at an exhaust velocity of $2\text{.}\text{00}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ . You may assume the gravitational force is negligible at the probe’s location.
$4\text{.}\text{16}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}$
Professional Application
Ion-propulsion rockets have been proposed for use in space. They employ atomic ionization techniques and nuclear energy sources to produce extremely high exhaust velocities, perhaps as great as $8\text{.}\text{00}\times {\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ . These techniques allow a much more favorable payload-to-fuel ratio. To illustrate this fact: (a) Calculate the increase in velocity of a 20,000-kg space probe that expels only 40.0-kg of its mass at the given exhaust velocity. (b) These engines are usually designed to produce a very small thrust for a very long time—the type of engine that might be useful on a trip to the outer planets, for example. Calculate the acceleration of such an engine if it expels $4\text{.}\text{50}\times {\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{kg/s}$ at the given velocity, assuming the acceleration due to gravity is negligible.
Derive the equation for the vertical acceleration of a rocket.
The force needed to give a small mass $\text{\Delta}m$ an acceleration ${a}_{\text{\Delta}m}$ is $F=\text{\Delta}{\text{ma}}_{\text{\Delta}m}$ . To accelerate this mass in the small time interval $\text{\Delta}t$ at a speed ${v}_{\text{e}}$ requires ${v}_{\text{e}}={a}_{\text{\Delta}m}\text{\Delta}t$ , so $F={v}_{\text{e}}\frac{\text{\Delta}m}{\text{\Delta}t}$ . By Newton’s third law, this force is equal in magnitude to the thrust force acting on the rocket, so ${F}_{\text{thrust}}={v}_{\text{e}}\frac{\text{\Delta}m}{\text{\Delta}t}$ , where all quantities are positive. Applying Newton’s second law to the rocket gives ${F}_{\text{thrust}}-\text{mg}=\text{ma}\Rightarrow a=\frac{{v}_{\text{e}}}{m}\frac{\text{\Delta}m}{\text{\Delta}t}-g$ , where $m$ is the mass of the rocket and unburnt fuel.
Professional Application
(a) Calculate the maximum rate at which a rocket can expel gases if its acceleration cannot exceed seven times that of gravity. The mass of the rocket just as it runs out of fuel is 75,000-kg, and its exhaust velocity is $2\text{.}\text{40}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ . Assume that the acceleration of gravity is the same as on Earth’s surface $\left(9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)$ . (b) Why might it be necessary to limit the acceleration of a rocket?
Given the following data for a fire extinguisher-toy wagon rocket experiment, calculate the average exhaust velocity of the gases expelled from the extinguisher. Starting from rest, the final velocity is 10.0 m/s. The total mass is initially 75.0 kg and is 70.0 kg after the extinguisher is fired.
How much of a single-stage rocket that is 100,000 kg can be anything but fuel if the rocket is to have a final speed of $8\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{km/s}$ , given that it expels gases at an exhaust velocity of $2\text{.}\text{20}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s?}$
$2\text{.}\text{63}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{kg}$
Professional Application
(a) A 5.00-kg squid initially at rest ejects 0.250-kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00-N frictional force opposing the squid’s movement. (b) How much energy is lost to work done against friction?
(a) 0.421 m/s away from the ejected fluid.
(b) $0\text{.237}\phantom{\rule{0.25em}{0ex}}\text{J}$ .
Unreasonable Results
Squids have been reported to jump from the ocean and travel $\text{30}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{m}$ (measured horizontally) before re-entering the water. (a) Calculate the initial speed of the squid if it leaves the water at an angle of $\text{20}\text{.}\mathrm{0\xba}$ , assuming negligible lift from the air and negligible air resistance. (b) The squid propels itself by squirting water. What fraction of its mass would it have to eject in order to achieve the speed found in the previous part? The water is ejected at $\text{12}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{m/s}$ ; gravitational force and friction are neglected. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent?
Construct Your Own Problem
Consider an astronaut in deep space cut free from her space ship and needing to get back to it. The astronaut has a few packages that she can throw away to move herself toward the ship. Construct a problem in which you calculate the time it takes her to get back by throwing all the packages at one time compared to throwing them one at a time. Among the things to be considered are the masses involved, the force she can exert on the packages through some distance, and the distance to the ship.
Construct Your Own Problem
Consider an artillery projectile striking armor plating. Construct a problem in which you find the force exerted by the projectile on the plate. Among the things to be considered are the mass and speed of the projectile and the distance over which its speed is reduced. Your instructor may also wish for you to consider the relative merits of depleted uranium versus lead projectiles based on the greater density of uranium.
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