453 Note_Set_1

453 Note_Set_1 - MAE 453 Intro to Space Flight Dr. Scott...

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MAE 453 – Intro to Space Flight Dr. Scott Ferguson Note Set 1 Page 1 of 13 Note Set 1 – Dynamics of Point Masses The purpose of this note set is to review the fundamental kinematics of point masses and basic vector operations. The topics covered in this note set provide the foundation capturing the orbital equations associated with the two- body problem. Introduction We’ll start with a review of the kinematics and dynamics of point masses ± Curvilinear motion of particles in three dimensions ± Force, mass, Newton’s inverse-square law of gravitation ± Second law of motion and angular momentum ± Motion relative to moving frames of reference Kinematics Tracking the motion of a particle requires a frame of reference that consists of: ± ± Given a reference frame, the position of a particle P at time t is defined by the position vector ) ( t r r ± ) ( t r r extends from the origin to the point itself ± In vector form: k t z j t y i t x t r ˆ ) ( ˆ ) ( ˆ ) ( ) ( + + = r (1-1) ± The magnitude of the distance from P to the origin is: 2 2 2 ) ( z y x r t r + + = = r (1-2) o What other mathematical operation can be used to find r?? The velocity ( v r ) and acceleration ( a r ) of the particle are the first and second time derivatives of the position vector: k t a j t a i t a k dt t dv j dt t dv i dt t dv t a k t v j t v i t v k dt t dz j dt t dy i dt t dx t v z y x z y x z y x ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ) ( + + = + + = + + = + + = r r (1-3)
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Dr. Scott Ferguson Note Set 1 Page 2 of 13 ± We will commonly represent the time derivatives with an overhead dot: r v a r r r = = So how does all of this help us? It lets us determine the path, or trajectory, or a particle: ± There are two types of motion dictated by the nature of the trajectory o o ± The velocity vector is tangent to the particle’s path t u v v ˆ = r ( 1 - 4 ) o The distance ds that P travels over the time interval dt is determined by: vdt ds = ± If we have v r , how can we find the components of the unit tangent t u ˆ in our frame of reference? Looking at the graphical representation of the velocity and acceleration vectors:
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This note was uploaded on 05/19/2010 for the course MAE 453 taught by Professor Mazzoleni,a during the Spring '08 term at N.C. State.

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453 Note_Set_1 - MAE 453 Intro to Space Flight Dr. Scott...

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