Physics of a Rocket's Trajectory

Physics of a Rocket's Trajectory

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We as humans have always been fascinated with the unknown.� We seek to conquer every frontier.� Today, the final frontier is space.� So, many people are very interested in rockets, the vehicle for conquering the final frontier.� Most people have a general idea of how rockets work, but very few have an understanding of the physics behind their flight, which scientists spent many years perfecting.

Rocket propulsion is not like many other kinds of propulsion that are based on the principle of a rotation based engine.� For example, a car engine produces rotational energy to turn the wheels of the car.� And, a airplane engine produces rotational energy to spin a turbine.� But, rocket propulsion is based on Newton�s Third Law, which says that for every action, there is an equal and opposite reaction.� So, rockets work by pushing fuel out the back, which in turn pushes the rocket forward.� The mass of the fuel pushed out the back of the rocket multiplied by the velocity of the fuel is equal to the mass of the rocket multiplied by the velocity of the rocket in the opposite direction.� Although there is always some energy loss in any type of engine, the rocket is propelled forward.



There are many forces that a rocket must overcome, especially during liftoff.� Newton�s second law says that force is equal to mass times acceleration (F=ma).� However, for a rocket the calculations are not that simple because the rocket�s mass is always changing as it burns up fuel.� So, we have to replace a new term with F, leading to

�where is a term for the thrust of the rocket and it is defined by R, the fuel consumption rate, and is the fuel�s exhaust speed relative to the rocket.� Also, we replace m with M and define M as the instantaneous mass of the rocket, including the unexpended fuel.

We also have to incorporate the other forces acting on the rocket, such as gravity and air resistance.� The force of gravity is equal to mg.� The force of air resistance is

�where C is the drag coefficient, is the air density, A is the cross-sectional area of the body perpendicular to the velocity, and v is the velocity.� By themselves, these formulas seem somewhat easy, but a rocket�s flight incorporates many variable forces that make the calculations much more difficult.� We have already examined the rocket�s upward force and how the changing mass makes the force vary.

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� But, the other forces on the rocket also vary.� The force of gravity is equal to Mg, but the mass of the rocket is always changing, and as the rocket travels higher and higher, gravity also changes.� And, the force of air resistance is



�but the coefficient of drag is not always constant, air density decreases with height above the earth, and velocity is always changing because the rocket is accelerating.� So, the calculations soon involve many integrals and become very difficult.

We can write the general equation for a rocket�s upward flight during liftoff as

We can apply this formula to any instant in the rocket�s path and find the net force fairly easily as long as we have all the information about the rocket at that instant.



By taking the integral of the formula we have for the flight of the rocket, we can analyze the rocket�s flight at any time t.� Although finding the value for a certain t would be fairly simple, the calculations become very time consuming when you have to perform them over and over again to analyze the whole flight of the rocket.� So, we can use a simple computer programming, such as Excel, so that the computer will do all the integrals for us.� By writing out the equations for conservation of energy, gravity, and air density and using the definition of the derivative, we come up with a final equation of:





We then start off with values of 0 for dt, v(t) and h(t) and plug our formula into Excel.� We then expand these formulas to include hundreds of calculations and graph the end result.



This graph demonstrates that as time increases, so do both velocity and height, which is to be expected.� Also, both increase slowly at first.� The velocity is increasing, so the rocket has some acceleration, which also causes the height to rise more quickly.

In order to completely break out of the earth�s gravitational field and continue on indefinitely, a particle has to exceed a certain initial velocity, called the escape velocity.� The escape velocity for a particle on earth�s surface is 40,320 km/h, or 11,200 m/s, which is very fast.� However, a rocket does not need to exceed that kind of a speed because the escape velocity is calculated for a particle with some initial velocity starting off from the ground and having no successive propulsion, such as a bullet shot from a gun.� A rocket, on the other hand can increase in velocity, keep a steady velocity, or at least keep from decreasing in velocity as much as a particle with only an initial velocity would.

Satellites placed into orbit never need to reach escape velocity either.� Instead, they are still acted on by gravity, which is why they are in orbit around the Earth.� The speed that they have to reach to achieve orbit is called orbital velocity, and it is a balance between the force of the gravitation pull of the earth and the rotational inertia of the satellite.� And, it varies according to the distance the satellite orbits from the Earth.� An approximate value for an orbit of 242 km (150 miles) is 27,359 km/h (17,000mph).

So, escape velocity is not all that important for rockets or their design for several reasons.� One is that most rockets never leave the Earth�s gravitation field, but instead fall into an orbit some distance from the earth.� Also, rockets have engines for propulsion and can increase in velocity if needed.� So, they do not simply have an initial velocity and can be propelled to whatever height is needed.

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