1. Relative Motion © Brendan Doheny, Applied Maths Facilitator,
St. Joseph’s Patrician College, Galway

2. Problem Types
There are two types of questions, each type is dealt with separately in this power-point application. The two types are as follows.
Type 1
Two particles moving, student is asked to find -:
(i) velocity of one particle relative to the other.
(ii) time taken to travel a certain distance.
the shortest distance between them in
subsequent motion.

3. Type 2
Boats moving in rivers – given speeds of river and the boat, we are asked to find-:
(i) the velocity of the boat relative to the river.
the time taken for the boat to cross the
river.
(iii) the quickest time to cross the river
the time taken to cross in the shortest
distance.

4. Background Knowledge required
Junior Cert maths students will be familiar with the concept of translations.
A vector, like a translation, has a direction and a magnitude associated with it.
The students will remember how to ‘translate’ an object based on direction and length(magnitude) .

5. Lets move the house below in a direction roughly south east and a distance of 5 units
After
Before

6. Scalars and Vectors
Scalar – physical quantity that is specified in terms of a single real number, or magnitude
eg. Length, temperature, mass, speed
Vector – physical quantity that is specified by both magnitude and direction
eg. Force, velocity, displacement, acceleration
The magnitude of a vector (length) is found by using the Pythagorean theorem:

7. RELATIVE VELOCITY 
As two objects, A & B move, the velocity of each object changes relatively, with respect to the other. The relative velocity of B with respect to A (also referred to as the velocity of B relative to A), is given by :

8. Example
If the velocity of a trawler is ms-1 and the velocity of a yacht is ms-1 .
Write down the-:
(a) velocity of the trawler relative to the yacht.
(b) velocity of the yacht relative to the trawler. 
End of Class 1

9. Exercises
If velocity of car is 12i + 5j m s-1 and the velocity of a truck is 2i – 5j ms-1, then find :
( a ) the speed and direction of the car.
( b ) the speed and direction of the truck.
( c ) the velocity of the car relative to the truck – represent your answer on a diagram.
( d ) the velocity of the car relative to the truck – represent your answer on a diagram.
{Solution: (a) 13 ms-1 , E22.62oN (b)5.385 m s-1 E21.8oS (c) 10i + 10j (d) -10i – 10j }
A car has a speed of 5m s-1 in a direction of East 60o North. Represent this on a diagram.
Express the velocity of the car as a vector.
{Solution: 5Cos 60o i Sin 60o j = 2.5i j }
A swan can swim at 10 m s-1 . If it has to swim to a point 40m north of its current position and then 30 m west – determine how long the journey will take.
Represent the velocity of the swan as a vector.
{Solution: -30i + 40j Journey time = 5 seconds}
A man can swim at 5m s-1 in a river , 100 m wide, that moves at 4m s-1 parallel to its banks. Show, in a diagram, the direction that he should swim in order to end up at a point directly across the river from where he started.
{Solution: W36.87oN}

10. Plotting the relative vector and calculating the time for interception
A trawler has velocity km/hr . A yacht has velocity km/hr. At 8am the trawler was 100 km west of the yacht. Calculate the closest they will be, in subsequent motion and the time this will occur.

11. STEPS TO SKETCH RELATIVE VECTOR
We place the yacht at the origin at 8am,
We locate the position of the trawler relative
to yacht at that time.
We then draw this vector from the
trawlers position
T km Y
Shortest distance

12. This has a magnitude of = 26.83 km/hr
making an angle of tanα with the horizontal i.e. tan-1(0.5). Direction E 26.56o S
So the shortest distance is
100 x Sinα = 100 x = km

13. The time taken is the length of the relative path divided by the speed along that path.
The length of the rel. path is the hypotenuse –
Cos α = => H = km.
The time taken was ( / 26.83) hours = hours 10 minutes.
T km Y
So they will be closest at ten minutes past noon.
Shortest distance
End of Class 2

14. Exercises
If velocity of car is m s-1 and the velocity of a truck is m s-1, and the truck is 1 km north of the car at 12 noon. Calculate the velocity of car relative to the truck, to the nearest ms-1 . Calculate the shortest distance between them subsequently.
Calculate the time they are closest together.
{Solution: / Distance = 1414 m // Time = 100 s }
A car has a speed of 24m s-1 in a direction of East 60o North. A man walks west at 1m s-1 and is 500m east of the car at 8am. . Calculate the velocity of car relative to the man.
Calculate the shortest distance between them subsequently and when it occurs.
{Solution: / Distance = m // Time = s }
A swan swims west at 12 m s-1 . A man swims north at 2 m s-1 . If the man is 20 m south of the swan – how close will they get and how long will it take?
{Solution: Distance = m / Time = 2.72 s }

15. 2008
Solution

16. 2008 ctd..
Solution

17. BOATS in rivers and PLANES in the sky!
This problem is slightly different as we are interested in the end resultant vector more so than the relative vector.
=>
In this case the resultant is VA as the resultant is defined as the sum of two vectors.

18. In the case of a boat crossing a river
=>
In this case the resultant is as the resultant is defined as the sum of two vectors.
Rules
If the shortest time to cross a river is required, then the relative velocity vector must be directly across the river.
If the shortest distance for a crossing is required, then the resultant velocity must be directly across the river i.e.

19. 2009
Solution
End of Class 3

20. Exercises
The velocity of a plane relative to the wind is m s-1 and the velocity of the wind is
m s-1. If the plane has to travel to a point 10km south of its present position, then calculate the time taken to arrive at its destination.
{Solution: seconds }
A boat has a speed of 24m s-1 in still water. A current has speed 3m s-1. If the river is 96 metres wide then, calculate the time taken for -:
( a ) for the boat to cross in the shortest time.
( b ) for the boat to cross in the shortest distance.
Represent these journeys on a vector diagram.
{Solution: (a) 4 seconds / (b) 4.03 seconds }