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 .

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.

Two dimensional relative position & velocity

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.

Relative position

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

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.

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?

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?

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?

i)

ii)

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

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

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)

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,

this week's promoted video

These are free to download and to share with others provided credit is shown.
Files cannot be altered in any way.
Under no circumstances is content to be used for commercial gain.