Relative Velocity

Claimed by Lily Masters (Fall 2016)

The motion of an object may look different when viewed from a different reference frame. This can be described by defining the relative velocity of the reference frame.

Relative Velocity

When an object is moving in a medium that is also moving, its velocity may be different depending on the location of the observer. For example, consider a boat moving through a flowing river. If the observer is aboard the boat, the velocity will be different than if the observer was standing by the side of the river. This can be more easily described through vector addition with one reference frame considered an intermediate reference frame:

[math]\displaystyle{ \vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC} }[/math]

This means that the velocity of A with respect to C is equal to the sum of the velocity of A with respect to B and the velocity of B with respect to C. In this case, B is the intermediate reference frame.

Examples

Airplane in Wind

An airplane is flying with a velocity of [math]\displaystyle{ \vec{v}_{PA} }[/math] relative to the air. The wind is moving with a velocity of [math]\displaystyle{ \vec{v}_{AG} }[/math] relative to an observer on the ground. The velocity of the plane relative to the ground can be found using vector addition:

[math]\displaystyle{ \vec{v}_{PG} = \vec{v}_{PA} + \vec{v}_{AG} }[/math]

Suppose the plane is moving with a velocity of [math]\displaystyle{ \left \langle {150,20,0} \right \rangle }[/math] km/h relative to the air. The wind is moving with a velocity of [math]\displaystyle{ \left \langle {-25,0,-10} \right \rangle }[/math] km/h relative to the ground. What is the velocity of the plane relative to the ground?

Answer:

[math]\displaystyle{ \vec{v}_{PG} = \left \langle {150,20,0} \right \rangle + \left \langle {-25,0,-10} \right \rangle }[/math]

[math]\displaystyle{ \vec{v}_{PG} = \left \langle {125,20,-10} \right \rangle }[/math]

Boat in Current

A boat is moving straight across a river with a velocity of [math]\displaystyle{ \vec{v}_{BW} }[/math] relative to the water. The river has a current flowing perpendicular to the boat which has a velocity of [math]\displaystyle{ \vec{v}_{WE} }[/math] relative to the earth. The velocity of the boat relative to the Earth can be found using vector addition and the bearing of the boat can be found using trig relations:

[math]\displaystyle{ \vec{v}_{BE} = \vec{v}_{BW} + \vec{v}_{WE} }[/math]

[math]\displaystyle{ \Theta = \arctan \frac{\left \Vert \vec{v}_{WE} \right \|}{\left \Vert \vec{v}_{BW} \right \|} }[/math]

Suppose the boat is moving straight across the river with a velocity of 37 m/s relative to the water and the current is moving downstream and perpendicular to the boat with a velocity of 4 m/s relative to the earth. What is the velocity of the boat relative to the earth and what is its bearing?

Answer:

[math]\displaystyle{ \vec{v}_{BE} = \left \langle {37,0,0} \right \rangle + \left \langle {0,-4,0} \right \rangle }[/math]

[math]\displaystyle{ \vec{v}_{BE} = \left \langle {37,-4,0} \right \rangle }[/math]

[math]\displaystyle{ \Theta = \arctan \frac{4}{37} }[/math]

[math]\displaystyle{ \Theta = 6.17018^\circ }[/math]

See also

• Velocity
• Speed and Velocity
• Derivation of Average Velocity
• 2-Dimensional Motion

External links

HyperPhysics: Relative Motion

References

HyperPhysics: Relative Motion