Up to this point the motion of a particle has been observed by a stationary observer. This is known as absolute motion. There can however be many cases where the motion of a particle could be so complex that it would be difficult to analyze by observing it from one stationary point. To help simplify a case like this, the particle can be observed from two or more frames of reference. An example of this could be analyzing the tip of a ships propeller. A fixed observer would see the ship itself moving, while another observer would analyze the propellers rotation. These two observations will then be superimposed vectorially in order to see the complete motion of the tip of the propeller.

**Position**

Let’s consider two particles that are moving in a straight line in different directions. These particles will be observed by a stationary observer at point O and there will also be a translating observer on particle A. The paths $s_A$ and $s_B$ are considered absolute positions since these paths are being observed by the stationary observer. The path $s_{B/A}$ however is considered a relative position that is observed by the translating observer on particle A. This can be expressed be by using vector addition as seen in the equation below.

**(Eq 1) ** $s_B=s_A+s_{B/A}$

**Velocity**

In order to find the velocity of the two particle being observed by a stationary observer and an observer on particle A the derivative of equation 1 would need to be found. This would result in the equation below.

**(Eq 2) **$v_B=\frac{ds_B}{dt}$, $v_A=\frac{ds_A}{dt}$, $v_{B/A}=\frac{ds_{B/A}}{dt}$

In the above equation $v_B$ and $v_A$ would be considered the absolute velocities, while $v_{B/A}$ would be considered the relative velocity. The velocities can also be related to each other through vector addition as seen in the equation below.

**(Eq 3)** $v_B=v_A+v_{B/A}$

**Acceleration**

Finally, in order to find the acceleration of two particles that are being observed by a stationary observer and an observer on particle A the derivative of the velocity equation would need to be taken, or the double derivative of the position equation would need to found. The equation below shows the derivative of equation 3.

**(Eq 4) **$a_B=\frac{dv_B}{dt}$, $a_A=\frac{dv_A}{dt}$, $a_{B/A}=\frac{dv_{B/A}}{dt}$

In the above equation $a_B$ and $a_A$ represent the absolute acceleration of the two particles, while $a_{B/A} represents the relative acceleration between particle A and particle B. The absolute accelerations and relative acceleration can be related to each other through vector addition as seen in the equation below.

**(Eq 5) **$a_B=a_A+a_{B/A}$

**Example**

What is the magnitude and direction of the relative velocity of particle A in respect to particle B?

**Solution**

Step one, use equation 3 and write the velocity components out in x and y coordinates.

$30i=(65cos(60)i+65sin(60)j)+V_{A/B}$

Solve for $V_{A/B}$

$V_{A/B}=(-2.5i-56.3j)\frac{m}{s}$

Find the magnitude of the velocity and angle.

$V_{A/B}=\sqrt{2.5^2+56.3^2}=56.4\frac{m}{s}$

$tan(θ)=\frac{-56.3}{-2.5}$, $θ=87.5^o$