Relative Velocity

All motion, and even time, is relative to an observer. A person that is walking forward at 4km/h on a ship will be seen as moving faster by an observer on the shore. This occurs because the speed of the ship is added to the person's speed when the situation is viewed relative to the shore.

If the ship and person are moving in different directions then a situation might occur when the person will think himself moving forward, while actually moing backwards.

When solving problems that involve relative velocity ( an object moving on another moving object etc. ) simply add the velocity of the two moving objects. Lets use the situation of a person walking on a ship to demonstrate relative velocity techniques.

Let the velocity of the person relative to the ship be Vps.
Let the velocity of the ship relative to the ground be Vsg.
Let the velocity of the person relative to the ground be Vpg.

Vpg = Vps + Vsg

The above formula can be used for just about any relative velocity situation. When you are given Vpg and Vsg and you need to find out
Vps rearrange the equation: Vps = Vpg - Vsg

It is important to recognize what a given velocity is relative to ( the ground, the water, the boat etc. ) and to make it clear in the variable's name. Also, the example above assumed that the water was still - if the water was moving you would have had to add its velocity to the velocity of the boat.


An airplane is flying with a constant velocity of 200km/h [S50°W], a wind of 15km/h [N4°W] is blowing and changing the plane's speed. What speed will the control tower, on the ground, see the plane moving at?

Let's define our variables: The plane produces enough energy output to make it move at 200km/h [S50°W] relative to the air so that,
Vpa = 200km/h [S50°W]

The velocity of the air relative to the ground is 15km/h [N4°W] so that,
Vag = 15km/h [N4°W]

And finally, the velocity of the plane relative to the ground is:

Vpg = Vag + Vpa
Vpg = 15 [N4°W] + 200 [S50°W]
Vpg = 15 Cos 4°[N] + 15 Sin 4°[W] + 200 Cos 50°[S] + 200 Sin 50°[W]
Vpg = -14.9[S] + 1[W] + 128.5[S] + 153.2[W]
Vpg = 154.2[W] + 113.6[S]

Lets find the speed,

|Vpg|² = 154.2² + 113.6²
|Vpg|² = 36682.6
|Vpg| = 191.5 m/s

And the angle of the plane's flight ( ß ),

Tan ß = 113.6 / 154.2
Tan ß = 0.7367
ß = 36°

The plane's velocity relative to the ground will be 190 km/h [W36°S].


Questions involving relative velocity are simple once you are comfortable with vectors. When solving a relative velocity question you have to take into account which velocity affects which and you also have to represent relativity in your variables ( let Vps represent the velocity of a person relative to a ship etc. ).