**APPLICATION OF VECTORS**

** Scalar and Vector Quantities **

Scalar quantities |
Vector quantities |

Are those physical quantities which have only magnitude |
Are the physical quantities which have both magnitude and direction |

Mass Length Area Density Speed etc |
Pressure Force Acceleration Velocity Momentum etc |

** Vector addition **

The addition of vector is done with the help of vector diagrams. The straight line is drawn to scale. The length of a line segment represents the magnitude of the

vector and the arrow represents the direction ie

When two or more vectors are added the sum is known as The resultant vector

Vectors can be added by mathematical methods using mathematical formula such as Pythagoras’ theorem, trigonometric Ratios etc

Also vectors can be added by graphical method

Adding vectors by **Graphical method ** o Choose a suitable scale and write it down on a graph paper

Pick a starting point and draw the first vector to scale direction stated (indicate the magnitude and direction)

Starting from the head of the first vector, draw the second vector to scale in the started direction until all given vectors finished

Draw the line to connect tail of the first drawn vector and the head of the last vector. This is called resultant vector

Measure the length of the resultant vector and convert to actual unit o Determine the direction of the resultant vector

** Example **

1. Suppose a man walks starting from point A, a distance of 20m due north and then walks 15m due east. Find his new position from A

Solution

Scale: 1cm represents 5cm

∴ From the fig above, AC (R) is 25 m at an angle of 36 51’ east of north

**The Triangle Method (Triangle law of vector addition) **

States that

“If two vectors are represented by two sides of a triangle in sequence, then the third closing side of the triangle ,drawn from the tail of the first vector to the head of the

second vector ,represents the resultant of the two vectors in both magnitude and direction “

OR

“If three vectors are in equilibrium and that two of the vectors are represented in magnitude and direction by two sides of a triangle, then the third side of the

triangle represents the resultant of the two forces”

**See the diagram below ****Example: **

1. A car is travelling due north at 60 km/hr .It turns and then travels due east at 80 km/hr .Find the magnitude and direction of the resultant velocity of the car Soln: By calculation method:

From: Pythagoras theorem: a2 + b2 = c2

a2 + b2 = c2 → R2 = 602 + 802 = 3400 +6400 = 10000 → 𝑅 = √10000 = 100

∴ 𝑇ℎ𝑒 𝑟𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 , 𝑹 𝒊𝒔 𝟏𝟎𝟎 km/h to the direction of N 53.1 E

**Individual Work –1:1 **

**<> **A brick is pulled by a force of 4N acting northward and another force of 3N acting north-east. Find the resultant of these two forces. ANS: The resultant is 6.5 N

**<> **Two forces, one 8 N and the other 6 N, are acting on a body. Given that the two forces are acting perpendicularly to each other, find the magnitude of the third force which would just counter the two forces. ANS = 10 N

**<> **A weight of 25 N is suspended from a beam by a string; what horizontal force must be applied to the weight to keep the string at an angle of 20 N to the vertical. What is the tension in the string? (ANS: FH =9.1N, TS = 26.6 N)

Parallelogram Method (Parallelogram law of vector addition) The law states that:

‘’If two vectors are represented by two adjacent sides of a parallelogram ,then the diagonal of the parallelogram through the common point represents the sum of the two vectors in both magnitude and direction’’ OR

“If two vectors are represented by the two sides given and they include angle between them, then resultant of the two vectors will be represented by the diagonal from their common point of a parallelogram formed by the two vectors” See the figure below

**Example1. **

Two forces of 20 N and 40 N acts at a point and the angle between them is 30. Find the resultant force and the angle it makes with the force of 20 (ANS: R = 58 N, at 20) Solution:

Scale: 1cm represents 5cm

From the graph above, R represents 11.6 Cm

Then: If 1cm = 5 N

11.6 cm =R (By crossing multiplication R = 58 N)

The resultant force is 58 N at an angle of 20

**Individual Task – 1:2 **

**<> **Two forces AB and AD of magnitude 40 N and 60 N respective are pulling a body on horizontal table. If the two forces makes an angle 30 between them, find the resultant force on the, body. (ANS: R = 96.7N)

**<> **Two ropes of 3 m and 6 m long are tied to a ceiling and their free ends are pulled by a force of 100 N as shown in the figure below. Find the tensions in each rope if they make angle 30° between them. (ANS: T1 = 34.5N, T2 = 69 N)

Find the resultant force when two forces act as shown in the figure below.

(ANS: RF = 10 N)

Find the resultant force, F, when two forces, 9 N and 15 N, act on an object with an angle of 600 between them. (ANS: FR = 21 N)

** Relative Velocity **

Relative velocity is the velocity of a body with respect to another moving body.

NB:

The relative velocity of a body with respect to a stationary observer is known as ‘’absolute relative velocity’’

Let the Velocity of one object be VA and that of another be VB denoted by symbol VAB.

If all objects are moving to the same direction, low speed will be experienced, therefore we minus two velocities of the moving bodies that is VAB = VA + (-VB) → VAB = VA – VB

If all objects are moving to the opposite direction, high speed will be experienced, therefore we plus two velocity of moving bodies VAB = VA – (-VB) → VAB = VA + VB NB:

Speed of an air plane may be observed by an observer on the ground to be increased by a TAIL WIND (VR = VA + VW) or reduced by HEAD WIND (VR = VA–VW). So the wind and the plane are both moving related to one another but the observer is stationary.

The speed of a boat in a river may also be observed by an observer at the river bank to be increased downstream or decrease up stream. Again the boat and the water are moving relative to one another but the observer is stationary.

Relative velocity also can be calculated by triangle method and by parallelogram method

**Example **

1. Two cars A and B are moving with velocities of 50 m/s and 30 m/s respectively. Find the relative velocity when they are in (a) The same velocity (b) Opposite velocity Soln:

Given: VA = 30 m/s, VB = 50 m/s

In the same velocity,

From: VAB = VA – VB → 𝑽𝑨𝑩 = 𝟓𝟎 − 𝟑𝟎 = 𝟐𝟎 𝒎/𝒔

In opposite direction

From: VAB = VA + VB → 𝑽𝑨𝑩 = 𝟓𝟎 + 𝟑𝟎 = 𝟖𝟎 𝒎/𝒔

** Individual Task – 1:3 **

A plane travelling at a velocity of 100 km/h to the South encounters a side wind blowing at 25 km/h to the West. What is its velocity relative to an observer on the ground? ANS: VR = 103.1km/h

Car A is moving with a velocity of 20 m/s while car B is moving with a velocity of 30 m/s. Calculate the velocity of car B relative to car A if:

They are moving in the same direction (ANS: VBA = 10 m/s)

They are moving in the opposite directions. (ANS: VBA = 50 m/s) 3. A Car is travelling at 60m/s due east and a lorry is travelling at 100m/s due north. What is the velocity of the car relative to the lorry? (ANS: 116.6 m/s at 310 NE)

An automobile A, travelling relative to the earth at 45km/h on a straight level road is ahead of motor cycle B travelling in the same direction at 90 km/h .What is the velocity of B relative to A? (ANS: VBA = 45km/h)

A passenger at the back of a train travelling at 15m/s relative to the earth , throws a hammer with a speed of 15 m/s in the opposite direction to the motion of the train .What is the velocity of the hammer relative to the earth (ANS: VHE = 0 m/s)

A boat heading due north crosses a wider river with a velocity of 36 m/s relative to the water .The river has a uniform velocity of 12 m/s due south

Determine the velocity of the boat with respect to an observer on the river bank (ANS 24 m/s )

If the river was flowing due east , determine the velocity of the boat with respect to an observer on the river bank (ANS 37.947m/s)

** Application of relative motion **

Is used in navigation to determine the actual velocity of vessels in moving water or in air

Is used to determine the velocities of stars and asteroids with respect to earth’s velocity

Used by structural engineers to design structures to avoid maximum deformation when earthquakes strike

Helps us to calculate the velocity of an object in a fluid

**Resolution of the Vector **

Is the process of splitting a vector into its components That is a horizontal and vertical components.

Consider the diagram below where the toy car is pulled at a certain angle but it seems to move horizontally due to horizontal force, not only that but also vertical force is formed

From the fig above: Horizontal vector = x and Vertical vector = y

The extract the triangle from the above diagram

Horizontal vector is given by the formula

𝑿 From: Cos θ = HC (X) = F Cos θ 𝑭 |

Vertical vector is given by the formula

𝒀 From: Sin θ = VC (Y) = F Sin θ 𝑭 |

** Individual Task – 1:4 **

A nail is being pulled using a string from a wall. The string forms an angle of 30° with the normal. If the force being used is 10 N, part of the force will tend to bend the nail while the other part will try to pull it out. Figure:

What is the magnitude of the force:

Tend to bend the nails? (ANS: F1 = 8.66 N)

Tend to pull the nails out? (ANS: F2 = 5.0 N)

A body is being acted on by two forces: F1 = 18 N acting at an angle of 25° and F2 = 30 N acting at 140° from due East. Find the resultant of the two forces, F, by separating the forces into x- and y- components. (ANS: RF = 27.70 N at an angle of 103.94 to west)

A trolley was pulled by a force of 60 N acting 30 to the horizontal .Find the vertical component of this force. ANS: VC = 30 N

A weight of 20 N rest on a plane inclined at 40 to the horizontal. What are the components of the weight parallel and perpendicular to the plane? (ANS: CParallel = 12.865 N, CPerpendicular = 15.32 N)

** Uses of Component Vectors**

- Adding many vectors
- Calculating a vector sum

**Class Assignment –1 **

1. The velocity of car B relative to car A is 8 m/s when the two cars are moving in the same direction and 28 m/s when the two cars are moving in opposite directions. Determine the velocity of each car

2. An aeroplane is taking off at a velocity of 20 m/s. Find the components of the plane’s velocity if the take – off angle is (a) 70 (b) 45 (c) 60

3. A river is flowing at a velocity of 2m/s due south .A person in a boat wants to move across the river at 10 m/s.

- In which direction should the person move?
- At what velocity should the person move the boat?

4. A box is being pulled on the floor using a string. The string makes an angle of 30 with the box as shown in the figure below

If the force being applied at the string is 200 N, find:

- The force which tends to lift the box
- The force which tends to pull the box forward

5. A car moves 5 km east 3 km south, 2 km west and 1 km north. Find the resultant displacement (ANS: 3.6 km 34 south of east)

6. A plane is flying at a velocity of 100 km/hr and wind is blowing at a velocity of 25 km/hr if the blowing wind is (a)head (b) tail

7. Find the resultant plane velocity relative of to an observer on the grounds 8. A motorboat traveling 5 m/s, East encounters a current traveling 2.5 m/s, North.

(a). What is the resultant velocity of the motor boat?

(b). If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?

(c).What distance downstream does the boat reach the opposite shore?

(ANS: (a) 5.59 m/s at 26.6 (b) 16 s (c) 40 m)

8. A man is walking inside a bus which is travelling at 56.2 km/hr. If the speed of the man relative to the ground is 55.8 km/h, is the man walking towards the front or the back?

9. A boat is travelling at 8.9 km/h. relative to the water in a river .The boat aims to straight for the opposite bank of the river which is 120.9 wide. If the speed of the water in the river is 2.9 km/h, how far downstream will the boat be when it reaches the opposite side?

10. A boat is travelling at 9.8 km/h relative to the water in a river wants to get a fishing camp that is 5.2 km upstream .If the speed of the water in the river is 6.0 km/h, how long will it take the boat to reach the camp?

11. A plane is flying at a velocity of 300 km/h, relative to the air towards 300 from due east .The plane is flying amidst a wind blowing at 85 km/h relative to the ground towards 225 from due east. What will be the velocity of the plane as observed on the control tower on the ground

12. An airplane is flying east at 200 km/h, which is its velocity relative to the air ,while a 100 km/h wind blows towards the north – east .What is its resultant velocity ? (ANS: V = 280km/h at 14.6)

13. A plane is flying due east with a velocity of 100 m/s when it encounters a wind moving at a velocity of 20 m/s . Find the resultant velocity of the plane if the direction of the wind is due

East (b) West (c) South (d) South – East

14. An aeroplane is taking off at a velocity of 20 m/s. Find the components of the plane’s velocity if the take – off angle is (a) 70 (b) 45 (c) 60

15. A River is flowing at a velocity of 2 m/s due south .A person in a boat wants to move across the river at 10 m/s

- In which direction should the person move?
- At what velocity should the person move the boat?

16. Two forces ,P and Q are applied on a small boat stuck in a shallow stream as shown below

Determine the magnitude and direction of the resultant of the two forces

17. The diagram in the figure below shows a block being pushed along a track .If a force of 20 N is applied in direction A at an angle of 60, what is the resolved part of the force in direction B ?

18. A Vehicle moving at a speed of 80km/hr emits smoke from its exhaust pipe in opposite direction at 50 km/hr with respect to the vehicle .Determine the speed of smoke with respect to the ground

19. When two motorbikes are approaching each other at constant speed ,the linear distance between them decreases at 7 km/hr .When moving in the same direction ,the linear distance between them decreases at 3 km/h Determine the velocity of each motorbike

20. A car is travelling due north at 45 km/hr .It turns and then travels due east at 72 km/hr .Find the magnitude and direction of the resultant velocity of the car

21. A Mass 3 kg hangs at the end of a string .Find the horizontal force needed to pull the mass sideways until the string is at 30 to the vertical .Find also the tension in the string

22. An air craft heads north – west at 320 km/hr relative to the wind .The wind velocity is 80km/hr from the south .Find the velocity of the aircraft relative to the ground

23. A deep sea diver dives at an angle of 30 with the surface of water and follows a straight – line path for a distance of 220 m. How far is the diver from the surface of water?

24. A velocity of magnitude 40 m/s is directed at an angle of 40 east of north .Represent this velocity on paper

25. A car travels 3 km due north ,then 5 km north–east .Represent these displacements graphically and determine the resultant displacement

26. Two forces , one of 12 N and another of 24 N, act on a body in such a way they make an angle of 30 with each other .Find the resultant of the two forces 28. A motorboat travelling 4 m/s ,East encounters a current travelling 3.0 m/s ,North (a) What is the resultant velocity of the motorboat?

27. If the width of the river is 80 m wide, then how much time does it take the boat to travel shore to shore? What distance downstream does the boat reach the opposite bank?

28. plane can travel with a speed of 80 mil/hr with respect to the air .Determine the resultant velocity of the plane if it encounters a

10 mil/hr headwind (c) 10 mil/hr crosswind

10 mil/hr tailwind (d) 60 mil/hr crosswind

Find the horizontal and vertical components of a force of 10 N acting at 30 to the vertical

29. A weight of 25 N is suspended from a beam by a string; what horizontal force must be applied to the weight to keep the string at an angle of 20 to the vertical .What is the tension in the string?

30. A car covered a displacement of 10 km due 30 ,then15 km due 120 and finally 8 km due 270 .Find the total displacement covered by the car

31. A man using a 70 kg garden roller on a level surface ,exerts a force of 200 N at 45 to the ground .Find the vertical force of the roller on the ground . If he pulls (ANS: 560 N) (b) If he pushes the roller (ANS: 840 N )

32. A plane mirror is approaching you at a speed of 10 m/s you can see your image in it. At what speed will your image approach you? (ANS: v = 20 m/s)

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