KINEMATICS OF PARTICLES

RELATIVE MOTION WITH RESPECT TO

TRANSLATING AXES

In the previous articles, we have described

particle motion using coordinates with respect to

fixed reference axes. The displacements,

velocities and accelerations so determined are

termed “absolute”.

It is not always possible or convenient to use a fixed

set of axes to describe or to measure motion. In

addition, there are many engineering problems for

which the analysis of motion is simplified by using

measurements made with respect to a moving

reference system. These measurements, when

combined with the absolute motion of the moving

coordinate system, enable us to determine the

absolute motion in question.

This approach is called the “ relative motion

analysis”.

In this article, we will confine our attention to moving

reference systems which translate but do not rotate.

Now let’s consider two particles A and B

which may have separate curvilinear

motions in a given plane or in parallel

planes; the positions of the particles at

any time with respect to fixed OXY

reference system are defined by and

.

Let’s attach the origin of a set of

translating (nonrotating) axes to

particle B and observe the motion of A

from our moving position on B.

Br

Ar

Fixed axis

Translating axis

The position vector of A as measured

relative to the frame x-y is ,

where the subscript notation “A/B”

means “A relative to B” or “A with

respect to B”.

jyixr BA

/

Fixed axes

Translating axes

The position of A is, therefore, determined by the vector

BABA rrr /

Fixed axes

Translating axes

We now differentiate this vector equation

once with respect to time to obtain velocities and

twice to obtain accelerations.

Here, the velocity which we observe A to have from

our position at B attached to the moving axes x-y is

This term is the velocity of A with respect to B.

BABArrr

/

BABABABA

rrrvvv//

jyixvr BABA

//

Acceleration is obtained as

Here, the acceleration which we observe A to have from our nonrotating position on B is .

This term is the acceleration of A with respect to B.

We note that the unit vectors and have zero derivatives because their directions as well as their magnitudes remain unchanged.

i

j

BABABABABABA vvvrrraaa /// ,

jyixavr BABABA

///

We can express the relative motion terms in whatever

coordinate system is convenient – rectangular, normal

and tangential or polar, and use their relevant

expressions.

1. The car A has a forward speed of 18 km/h and is

accelerating at 3 m/s2. Determine the velocity and acceleration

of the car relative to observer B, who rides in a nonrotating

chair on the Ferris wheel. The angular rate = 3 rev/min of

the Ferris wheel is constant.

4. Car A is traveling along a

circular curve of 60 m radius at a

constant speed of 54 km/h. When

A passes the position shown, car B

is 30 m from the intersection,

traveling with a speed of 72 km/h

and accelerating at the rate of 1.5

m/s2. Determine the velocity and

acceleration which A appears to

have when observed by an

occupant of B at this instant. Also

determine r, , , , and for

this instant.

r

60 m

30o

30 m A

B

r

r