Analysing Relative Motion – Learn

Analysing Relative Motion

As you should recall from a previous section, relative velocity is the velocity of an object relative to another moving object. In this course, three main situations are described:

A boat on a flowing river relative to the bank
Two moving cars
An aeroplane in a cross wind relative to the ground

Lets analyse each of these situations using a diagram:

A boat on a flowing river relative to the bank:

For an observer on the bank, there are two factors which may contribute to the motion of the boat: the boats motion relative to the water and also the flowing water relative to the bank. In the following diagram, the boat is moving to the right (relative to the water, v_b). The water is flowing down (relative to the bank, v_w). The velocity of the boat relative to the observer on the bank is v_bo:

The velocity of the boat relative to an observer on the bank could be calculated as:

The velocity of the boat relative to an observer on the bank: v_bo = v_b + v_w

Two moving cars

The diagram shows two cars: a blue car moving down with a velocity relative to the ground of v_b and a red car moving to the right with a velocity relative to the ground of v_r.

The relative velocity of either car could calculated:

The velocity of the red car relative to the blue car: v_rb = v_r – v_b = v_r + (-v_b)
The velocity of the blue car relative to the blue car: v_br = v_b – v_r = v_b + (-v_r)

An aeroplane in a cross wind relative to the ground

For an observer on the ground, there are two factors which may contribute to the motion of the plane: the planes motion relative to the wind and also the wind relative to the ground. In the following diagram, the plane is moving to the right (relative to the wind, v_p). The wind is blowing down (relative to the ground, v_w). The velocity of the plane relative to the observer on the ground is v_pg:

The velocity of the plane relative to an observer on the ground could be calculated:

The velocity of the plane relative to an observer on the ground: v_pg = v_p + v_w

Things to look out for:

Some problems may only require an analysis in one dimension. For example, the boat and tide, plane and wind, or the two cars, travelling in the same or opposite directions
Many of the problems will be in 2-D and will require a 2-D vector analysis in order to solve
Some problems may involve determining another variable like the wind or water speed. This will require the above calculations to be rearranged.
Any problems involving 2-D will require some trigonometry to be solved.

Drawing diagrams can often assist in visualising and solving these types of problems!

Example 1:

A plane is flying north relative to the wind at 40 m/s. The plane encounters an easterly cross wind of 14 m/s. Calculate the velocity of the plane relative to an observer on the ground:

Answer:

Start with a diagram:

v_pg = v_p + v_w– this requires trigonometry using the above diagram.

$({ v }_{ pg })^{ 2 }={ { v }_{ p } }^{ 2 }+{ { v }_{ w } }^{ 2 }$

$({ v }_{ pg })^{ 2 }={ 40 }^{ 2 }+14^{ 2 }$

$({ v }_{ pg })^{ 2 }=1600+196 }$

${ v }_{ pg }=\sqrt { 1796 }$

${ v }_{ pg }=42.4$ m/s

direction:

$\tan { \theta } =\cfrac { 14 }{ 40 }$

$\theta \; =\tan ^{ -1 }{ \cfrac { 14 }{ 40 } }$

$\theta \; ={ 19.3 }^{ \circ }$

v_pg = 42.4 m/s @ N19.3°E

Example 2:

A marshall on the river bank observes a boat travelling at 5.5 m/s in a direction of N30°E. The tidal water is travelling at 1.5 m/s east. Assuming the boat is travelling north relative to the bank, what is the velocity of the boat relative to the water?

diagram:

Looking at the diagram, we can see that is a trig problem to solve v_b by rearranging pythgoras’ theorem:

$({ v }_{ bo })^{ 2 }={ { v }_{ b } }^{ 2 }+{ { v }_{ w } }^{ 2 }$

${ { v }_{ b } }^{ 2 }=({ v }_{ bo })^{ 2 }-{ { v }_{ w } }^{ 2 }$

${ { v }_{ b } }^{ 2 }=5.5^{ 2 }-{ 1.5 }^{ 2 }$

${ { v }_{ b } }=\sqrt { 28 }$

${ { v }_{ b } }=5.3$ m/s north