**Home** >> Mechanics, 2D motion, relative motion

**MECHANICS**

**2D Motion**

**Relative Motion**

velocity one dimension |

__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 **x**_{A} from **O**, and a velocity **V**_{A} along the **x-axis**.

The displacement **x**** _{A}** is a function of time

**t**.

Particle **B** has a displacement **x**** _{B}** from

**O**, and a velocity

**V**

_{B}along the

**x-axis**.

The displacement **x**** _{B}** is also a function of time

**t**.

The velocity **V**_{B} relative to velocity **V**_{A} is written,

_{B}**V**_{A} = **V**_{B} - **V**_{A}

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 **r**_{A} from **O**, and a velocity **V**_{A} along the **x-axis**.

The displacement** r _{A}** is a function of time

**t**.

Particle **A** has a displacement **r**_{B} from **O**, and a velocity **V**_{B} along the **x-axis**.

The displacement **r**** _{B}** 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**, **r**** _{B-A}** .

The position vector **r**** _{B-A}** can be written as,

_{B}r_{A} = r_{B}- **r**_{A}

__Relative velocity__

Similarly, at time **t** the velocity vector **V**_{B} relative to velocity vector **V**_{A} can be written,

_{B}**V**_{A} = **V**_{B} - **V**_{A}

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 (9**i** - 2**j**) ms^{-1} and the velocity of another particle **Q** is

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

__Example #2__

A particle **P** has a velocity (4**i** + 3**j**) ms^{-1}. If a second particle **Q** has a relative velocity to **P** of (2**i** - 3**j**), 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 (4**i** + 3**j**) km, with velocity vector (3**i** - **j**) km hr ^{-1}.

**Q** has position vector (8**i** + **j**) km, with velocity vector (2**i** + 2**j**) 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 **a _{A}** and

**a**are the acceleration vectors at

_{B}**A**and

**B**at time

**t**,

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

__this week's promoted video__

[ About ] [ FAQ ] [ Links ] [ Terms & Conditions ] [ Privacy ] [ Site Map ] [ Contact ]