MECHANICS - 2D Motion

 

Relative Motion

 

 

velocity one dimension

velocity 2D

acceleration 2D

 

 

 

One dimensional relative velocity(in a line)

 

Consider two particles A and B at instant t positioned along the x-axis from point O.

 

Particle A has a displacement xA from O, and a velocity VA along the x-axis.

The displacement xA is a function of time t .

 

Particle B has a displacement xB from O, and a velocity VB along the x-axis.

The displacement xB is also a function of time t .

 

 

one dimensional relative velocity

 

 

The velocity VB relative to velocity VA is written,

 

 

BVA = VB - VA

 

 

This can be expressed in terms of the derivative of the displacement with respect to time.

 

 

velocity in terms of displacement

 

 

relative velocity in terms of displacement derivative

 

 

 

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Two dimensional relative position & velocity

 

 

2D velocity and displacement

 

 

Particle A has a displacement rA from O, and a velocity VA along the x-axis.

The displacement rA is a function of time t .

 

Particle A has a displacement rB from O, and a velocity VB along the x-axis.

The displacement rB is also a function of time t.

 

 

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Relative position

 

 

2D relative displacement

 

 

The position of B relative to A at time t is given by the position vector from O, rB-A .

 

The position vector rB-A can be written as,

 

BrA = rB- rA

 

 

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Relative velocity

 

 

2D relative velocity using vectors

 

 

Similarly, at time t the velocity vector VB relative to velocity vector VA can be written,

 

BVA = VB - VA

 

This can be expressed in terms of the derivative of the displacement with respect to time.

 

 

relative velocity in terms of derivative

 

 

 

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Example #1

 

If the velocity of a particle P is (9i - 2j) ms-1 and the velocity of another particle Q is

(3i - 8j) ms-1 , what is the velocity of particle P relative to Q?

 

 

relative motion problem#01

 

 

Example #2

 

A particle P has a velocity (4i + 3j) ms-1. If a second particle Q has a relative velocity to P of     (2i - 3j), what is the velocity of Q?

 

 

relative motion problem#2

 

 

 

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Example #3

 

A radar station at O tracks two ships P & Q at 0900hours (t=0) .

 

P has position vector (4i + 3j) km, with velocity vector (3i - j) km hr -1.
Q has position vector (8i + j) km, with velocity vector (2i + 2j) km hr -1.

 

i) What is the displacement of P relative to Q at 0900 hours? (ie distance between ships). Answer to 2 d.p.

 

ii) Write an expression for the displacement of P relative to Q in terms of time t .

 

iii) Hence calculate the displacement of P relative to Q at 1500 hours.

 

iv) At what time are the two ships closest approach and what is the distance between them at this time?

 

 

relative motion problem#3a

 

 

i)

 

relative motion problem 3i answer

 

 

ii)

 

relative motion problem#3 definitions

 

 

relative motion problem#3rpt

 

 

relative motion problem#3rqt

 

 

therefore the displacement of P relative to Q is given by,

 

 

relative motion problem#3ii displacement in terms of t

 

 

iii) using the result above for 1500 hours( t = 6 )

 

relative motion problem #3iii

 

 

iv) Closest approach is when the position vector of P is at right angles to the reference vector.

The 'reference vector' is the first part of the vector equation for r .

The position vector gives the point P at time t along the straight line described by the vector equation.

(solution to follow)

 

 

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Two dimensional relative acceleration

 

Similarly, if aA and aB are the acceleration vectors at A and B at time t,

 

then the acceleration of B relative to A is given by,

 

 

2D relative acceleration

 

 

 

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