,
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 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 a function of time t .

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

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

 

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

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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