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  • State Newton’s third law of motion.
  • Explain the principle involved in propulsion of rockets and jet engines.
  • Derive an expression for the acceleration of the rocket.
  • Discuss the factors that affect the rocket’s acceleration.
  • Describe the function of a space shuttle.

Rockets range in size from fireworks so small that ordinary people use them to immense Saturn Vs that once propelled massive payloads toward the Moon. The propulsion of all rockets, jet engines, deflating balloons, and even squids and octopuses is explained by the same physical principle—Newton’s third law of motion. Matter is forcefully ejected from a system, producing an equal and opposite reaction on what remains. Another common example is the recoil of a gun. The gun exerts a force on a bullet to accelerate it and consequently experiences an equal and opposite force, causing the gun’s recoil or kick.

Making connections: take-home experiment—propulsion of a balloon

Hold a balloon and fill it with air. Then, let the balloon go. In which direction does the air come out of the balloon and in which direction does the balloon get propelled? If you fill the balloon with water and then let the balloon go, does the balloon’s direction change? Explain your answer.

[link] shows a rocket accelerating straight up. In part (a), the rocket has a mass m size 12{m} {} and a velocity v size 12{v} {} relative to Earth, and hence a momentum mv size 12{ ital "mv"} {} . In part (b), a time Δ t size 12{Δt} {} has elapsed in which the rocket has ejected a mass Δ m size 12{} {} of hot gas at a velocity v e size 12{v rSub { size 8{e} } } {} relative to the rocket. The remainder of the mass m Δ m size 12{ left (m - right )} {} now has a greater velocity v + Δ v size 12{ left (v+Δv right )} {} . The momentum of the entire system (rocket plus expelled gas) has actually decreased because the force of gravity has acted for a time Δ t size 12{Δt} {} , producing a negative impulse Δ p = mg Δ t size 12{Δ`p= - ital "mg"Δ`t} {} . (Remember that impulse is the net external force on a system multiplied by the time it acts, and it equals the change in momentum of the system.) So, the center of mass of the system is in free fall but, by rapidly expelling mass, part of the system can accelerate upward. It is a commonly held misconception that the rocket exhaust pushes on the ground. If we consider thrust; that is, the force exerted on the rocket by the exhaust gases, then a rocket’s thrust is greater in outer space than in the atmosphere or on the launch pad. In fact, gases are easier to expel into a vacuum.

By calculating the change in momentum for the entire system over Δ t size 12{Δ`t} {} , and equating this change to the impulse, the following expression can be shown to be a good approximation for the acceleration of the rocket.

a = v e m Δ m Δ t g size 12{a= { {v"" lSub { size 8{e} } } over {m} } { {Δm} over {Δt} } - g} {}

“The rocket” is that part of the system remaining after the gas is ejected, and g size 12{g} {} is the acceleration due to gravity.

Acceleration of a rocket

Acceleration of a rocket is

a = v e m Δ m Δ t g , size 12{a= { {v"" lSub { size 8{e} } } over {m} } { {Δm} over {Δt} } - g,} {}

where a size 12{a} {} is the acceleration of the rocket, v e size 12{v rSub { size 8{e} } } {} is the escape velocity, m size 12{m} {} is the mass of the rocket, Δ m size 12{Δm} {} is the mass of the ejected gas, and Δ t size 12{Δt} {} is the time in which the gas is ejected.

Picture a shows a rocket launched into space. It moves upward with velocity v in time t and the burning of fuel is also shown. After time t plus delta t the mass of fuel decreases by delta m and hence the velocity of the rocket increases to v plus delta v. The free body diagram shows the weight W of the rocket downward, reaction force upward and the resultant velocity upward too.
(a) This rocket has a mass m size 12{m} {} and an upward velocity v size 12{v} {} . The net external force on the system is mg size 12{ size 11{ - ital "mg"}} {} , if air resistance is neglected. (b) A time Δ t size 12{Δ`t} {} later the system has two main parts, the ejected gas and the remainder of the rocket. The reaction force on the rocket is what overcomes the gravitational force and accelerates it upward.

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Source:  OpenStax, Introduction to applied math and physics. OpenStax CNX. Oct 04, 2012 Download for free at http://cnx.org/content/col11426/1.3
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