FURTHER MATHEMATICS SS 1 SECOND TERM
WEEK 6 DATE(S)……………………………………….
SUBJECT: FURTHER MATHEMATICS
TOPIC: RELATIVE VELOCITY
- Displacement/Relative displacement
- Relative velocity
- Motion of word problem
SUB – TOPIC: Relative velocity
Every object or occurrence (be it real or abstract) is relative to some other object, occurrence. If a ship has a speed of 10m/s in still water, then we are referring to its speed relative to earth.
Note the following points when solving relative velocity problems:
- Identify the parameters in the problem i.e. relative velocity, magnitude, direction.
- Draw the displacement diagram.
- Write the relative velocity equation, making use of the Domino rule; e.g. AVB =AVE+EVB i.e. the velocity of A relative to B is equal to the velocity of A relative to the earth plus velocity of earth relative to B. This rule can be extended to any number of terms.
Example 1: Ship A is sailing due East at 12m/s and ship B is sailing due North at 16m/s. Find the velocity of ship B relative to ship A.
AVE 12m/s East
BVE 16m/s North
Since BVE = BVA + AVE and we seek BVA, then BVA = BVE – AVE
Applying Pythagoras’ theorem to the triangle above gives:
BVA = , and makes direction with BVE
Example 2: If ship A is moving North – east at 15m/s and a second ship B appears to an observer from A to be moving east at 7m/s. Find the actual velocity of B.
Solution: Magnitude Direction
AVE 15m/s North – East
BVA 7m/s East
Since BVE = BVA + AVE
Applying cosine rule, (BVE)2 = 72 + 152 – 2 × 7 × 15
Therefore, BVE =20.6m/s
Applying sine rule,
The direction is about N590E
- One ship is sailing due east at 24km/h and another ship is sailing due north at 32km/h. Find the velocity of the second ship relative to the first.
- If a ship A is moving North – East at 12km/h and an engine boat B appear to an observer from A to be moving south at 10km/h. Find the actual velocity of ship B.
SUB – TOPIC: Displacement
Displacement is the distance moved by a point in a specified direction. It is a vector quantity, and therefore has both magnitude and direction. If I walk 4km East and 3km South, then I have been displaced 5km from my starting point, despite the fact that I have walked 7km altogether.
The displacement of a point moving from the point A to the point B can be with respect to the origin O , can be represented by the vector AB.
If the position vectors of A and B are a and b respectively, then:
AB = AO + OB
= – OA + OB
= – a + b
We use the diagram above to consider the displacement vector of P relative to R. Let initial position of P be PO and that of R be RO. Let the position of P after time t be Pt, and that of R after time t be Rt. Let the velocity of P be VP and the velocity of R be VR. After a time t, P will be at Pt so
OPt = OPO + tVP
Similarly, ORt = ORO + tVR
Considering the vector polygon above,
RtPt = RtRo+ ROPO + POPt
= – tVR + ROPO + tVp
= ROPO + tVP – tVR
= ROPO + t(VP – VR)
Hence: RtPt = ROPO+ t(VP – VR)
i.e. displacement = time × Velocity ( of P relative to R)
Example 1: An aircraft A flies at 800km/h due west. Another aircraft B, which is 20km due north of A sets off to pursue A at 1200km/h. Find:
- The course of aircraft
- The time of interception of the aircraft A by the aircraft B
Solution: the velocity diagram is shown below
Let BVE = velocity of B relative to earth
AVE = velocity of A relative to earth
BVA = velocity of B relative to A
Then, = 1200
to the nearest whole number.
Hence, the course (direction) of aircraft B is (i.e. .
(ii) to get the time of interception, we consider the displacement diagram
Let the distance along the course be , then
Solving for we get
Example 2: A boat A sails due west at 15km/h. Another boat B, initially 40km due north of the first, sail at 10km/h on the bearing. Find:
- The shortest distance between the two boats
- The time when they are at this distance.
Solution: consider the velocity diagram below
VA – VB
Where VA = velocity of boat A
VB = velocity of boat B
VA – VB = velocity of boat A relative to boat B
Let . Then using the cosine rule;
= 225 + 100 –
Let then using the sine rule,
Let be the initial position of boat A
be the initial position of boat B
be the position of boat B at time t
Consider the displacement diagram below
The shortest distance between the two boats is given by the length of the perpendicular from to Let the perpendicular from to be
(ii) Let t be the time that they are at the shortest distance apart, then
- A helicopter travels at 100km/h in still air. If the wind is blowing from the west at 40km/h, how long will it take the helicopter to reach a place 250km off to the south – west?
- A trawler A is 40km west of another traveler B. A set off at 20km/h on a course of 0600. If the trawler B can travel at 25km/h, what course should trawler B take to intercept trawler A?
SUB – TOPIC: Acceleration/motion of Aircraft and Ship
Acceleration is the rate of change of velocity with time. It is a vector quantity. If the magnitudes of accelerations represented by the adjacent sides of a parallelogram are and , with the angle between them being β, then the magnitude of the resultant acceleration denoted by is given by:
. If = 900, then
When an aircraft moves through the air, it is affected by the motion of the air. The diagram below illustrates the relationship between the course/airspeed, direction of wind/wind speed and track/ground speed.
Wind direction/wind speed
The angle of drift is . The motion of a ship in water is similar to the motion of an aircraft in the air.A similar vector diagram for the motion of a ship in water is shown below.
Speed of ship
Example 1: an aircraft flies with an air speed of 160km/h from its base P to a point Q which is 65km East and 130km North of P. The wind is blowing from the west at a speed of 32km/h. find the:
- Direction in which it is headed
- Time, to the nearest minuet, taken on its journey from P to Q
α = 63.430
It follows that
Wind speed = 32km/h
Air speed = 160km/h
Using sine rule in
Hence the direction in which the aircraft is headed is N16.260E
Let the ground speed be V, then using sine rule,
We get From above thus;
Consider the displacement diagram PRQ below:
In , let y be the distance from P to Q, then
Let the time taken to fly from P to Q be t
Since time =
= 0.8462h or 51mins.
- An aircraft flew out of Benin Airport on a bearing of 3150 and at an air speed of 250km/h. if the speed and direction of the wind were 50km/h and 0750, calculate, correct to the nearest whole number:
- The ground speed
- The angle of drift
- The track
- A ship P is sailing due east with a speed of 4µm/s. The speed of the second ship Q relative to P is µ m/s south – west. Find in m/s, the speed of Q.
- A ship P sails due east with speed 4tm/s. A second ship Q sails south – west with a speed of m/s relative to Q. Find in m/s, the speed of Q.
- Two particles A and B have velocities and respectively. Find the velocity of B relative to A.
- An aircraft has airspeed of 96km/h and flies in the direction S300W. A wind blows 40km/h from the direction N400W. Calculate the ground speed of the aircraft.
- The airspeed of an aircraft is 250km/h and the wind blows from the direction S450W with a speed of 40km/h. the pilot of the aircraft wishes to travel due north. In what direction must he steer the aircraft?
- Two cars A and B are travelling in perpendicular directions along straight roads which intersect at O. A moves with a velocity of 30km/h and B moves with a velocity of 50km/h. Find the velocity of A relative to B and the direction of car A.
- The position vector of a point P is and the point Q has position vector where are unit vectors in direction of East and North respectively. Calculate:
- The distance PQ, correct to the nearest 0.1m
- The bearing of Q from P, correct to the nearest degree.
- A and B are two airports with B 400km due east of A. An aircraft whose speed in still air is 200km/h flies directly from A to B through a wind blowing from the northeast at 40km/h. Find the speed of the aircraft relative to the ground. On another day, the aircraft set out from A to B in still air. At the same time another aircraft starts to fly north from B at 150km/h. Determine the shortest distance between the two aircrafts and the time taken, correct to the nearest minute, before they are closest to each other.
- GROUND SPEED
- ANGLE OF DRIFT
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