The Vectors How to add and find the res
- 2. • Add vectors (force or velocity) graphically Use a scale diagram to find the resultant graphically for vectors along the same line or perpendicular to each other. The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector. According to the parallelogram law of vector addition if two vectors act along two adjacent sides of a parallelogram(having magnitude equal to the length of the sides) both pointing away from the common vertex, then the resultant is represented by the diagonal of the parallelogram passing through the same common vertex.
- 3. • Add vectors (force or velocity) mathematically 1. Add or subtract vectors if they are acting along the same line 2. For perpendicular vectors, use Pythagoras theorem to find the resultant mathematically Finding the resultant vector Apply the graphical or mathematical methods of adding vectors to unfamiliar situations
- 4. SCALAR A SCALAR quantity is any quantity in physics that has MAGNITUDE ONLY Number value with units Scalar Example Magnitude Speed 35 m/s Distance 25 meters Age 16 years
- 5. VECTOR A VECTOR quantity is any quantity in physics that has BOTH MAGNITUDE and DIRECTION Vector Example Magnitude and Direction Velocity 35 m/s, North Acceleration 10 m/s2 , South Displacement 20 m, East
- 6. Vector quantities can be identified by bold type with an arrow above the symbol. V = 23 m/s NE
- 7. Vectors are represented by drawing arrows
- 8. The length and direction of a vector should be drawn to a reasonable scale size and show its magnitude 20 km 10 km
- 9. Representing Vectors Vectors are represented by an arrow. • The arrowhead indicates the direction of the vector • The length of the arrow represents the magnitude The two force vectors acting on the object have both a direction and a magnitude.
- 11. Step 1: Draw the first tennis ball and its velocity vector • Measure the 45° angle with a protractor Step 2: Draw the second tennis ball and its velocity vector • The second ball has a speed of 10 m/s, so the arrow will be twice as long
- 12. • Learning Outcomes Key Concept 4: Vectors • Determine, by calculation or graphically, the resultant of two vectors at right angles, limited to forces or velocities only • For two vectors and , that make a right angle to each other, the Pythagorean theorem is a 𝐴 𝐵 useful method for determining the resultant ( ) of the two vectors. 𝑅 Add vectors (force or velocity) graphically Add vectors (force or velocity) mathematically
- 13. Resultant A resultant vector is the sum of two or more vectors. The resultant of two or more vectors that do not act along the same line can be found by drawing a vector triangle or by calculation. The resultant vector 𝑅 is defined such that 𝐴 + 𝐵 = 𝑅
- 14. Resultant The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
- 15. •Parallelogram law of vector addition • According to the parallelogram law of vector addition if two vectors act along two adjacent sides of a parallelogram(having magnitude equal to the length of the sides) both pointing away from the common vertex, then the resultant is represented by the diagonal of the parallelogram passing through the same common vertex.
- 16. VECTOR APPLICATION • ADDITION: When two (2) vectors point in the SAME direction, simply add them together. • When vectors are added together they should be drawn head to tail to determine the resultant or sum vector. • The resultant goes from tail of A to head of B.
- 17. A man walks 46.5 m east, then another 20 m east. Calculate his displacement relative to where he started. •Let’s Practice 66.5 m, E 46.5 m, E + 20 m, E
- 18. •VECTOR APPLICATION SUBTRACTION: When two (2) vectors point in the OPPOSITE direction, simply subtract them.
- 19. Let’s Practice some more…. A man walks 46.5 m east, then another 20 m west. Calculate his displacement relative to where he started. 26.5 m, E 46.5 m, E - 20 m, W
- 20. Graphical Method Aligning vectors head to tail and then drawing the resultant from the tail of the first to the head of the last.
- 21. Graphical Vector Addition A + B Step 1 – Draw a start point Step 2 – Decide on a scale Step 3 – Draw Vector A to scale Step 4 – Vector B’s tail begin at Vector A’s head. Draw Vector B to scale. Step 5 – Draw a line connecting the initial start point to the head of B. This is the resultant.
- 23. Forces as Vectors Net Force • Vector diagrams include arrows in a particular direction which represent the different forces on an object. • The size of the arrow corresponds to the size of the force • Net, or resultant, forces can be calculated by adding or subtracting all of the forces acting on the object • Forces working in opposite directions are subtracted from each other • Forces working in the same direction are added together • If the forces acting in opposite directions are equal in size, then there will be no resultant force – the forces are said to be balanced
- 24. NON CO-LINEAR VECTORS When two (2) vectors are PERPENDICULAR to each other, you must use the PYTHAGOREAN THEOREM
- 25. Let’s Practice A man travels 120 km east then 160 km north. Calculate his resultant displacement. c2 a2 b2 c a2 b2 c resultant 120 2 160 2 c 200km VERTICAL COMPONENT FINIS H 120 km, E 160 km, N the hypotenuse is called the RESULTANT HORIZONTAL COMPONENT STAR T
- 26. Step 1: Draw a vector diagram
- 27. Calculating Vectors Graphically (Parallelogram Method) Vectors at right angles to one another can be combined into one resultant vector. • The resultant vector will have the same effect as the two original ones • To calculate vectors graphically means carefully producing a scale drawing with all lengths and angles correct • This should be done using a sharp pencil, ruler and protractor Follow these steps to carry out calculations with vectors on graphs • Choose a scale which fits the page • For example, use 1 cm = 10 m or 1 cm = 1 N, so that the diagram is around 10 cm high • Draw the vectors at right angles to one another (tail to tail) • Complete the resulting parallelogram. The resultant vector is the diagonal of the parallelogram. • Draw the resultant vector diagonally from the origin • Carefully measure the length of the resultant vector • Use the scale factor to calculate the magnitude • Use the protractor to measure the angle Example: Find the resultant of vectors a and b below
- 28. Step 1: Draw a vector diagram Step 2: Calculate the magnitude of the resultant vector using Pythagoras’ theorem Step 3: Calculate the direction of the resultant vector using trigonometry Step 4: State the final answer complete with direction Resultant vector = 12 km 59° east and upwards from the horizontal
- 29. •WHAT ABOUT DIRECTION? In the example, DISPLACEMENT is asked for and since it is a VECTOR quantity, we need to report its direction. N S E W N of E E of N S of W W of S N of W W of N S of E E of S NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. N of E
- 30. •Directions • There is a difference between Northwest and West of North
- 31. •NEED A VALUE – ANGLE! Just putting N of E is not good enough (how far north of east ?). We need to find a numeric value for the direction. 160 km, N 120 km, E To find the value of the angle we use a Trig function called TANGENT. Tan opposite side adjacent side 160 120 1.333 Tan 1 (1.333) 53.1o N of E q 200 km So the COMPLETE final answer is : 200 km, 53.1 degrees North of East
- 32. •What are your missing components? Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? E m C H opp N m C V adj hyp opp hyp adj hypotenuse side opposite hypotenuse side adjacent , 47 . 27 25 sin 65 . . , 91 . 58 25 cos 65 . . sin cos sine cosine 65 m 25˚ H.C. = ? V.C = ? The goal: ALWAYS MAKE A RIGHT TRIANGLE! To solve for components, we often use the trig functions sine and cosine.
- 33. •Example A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 3 . 31 ) 6087 . 0 ( 6087 . 23 14 93 . 26 23 14 1 2 2 Tan Tan m R 35 m, E 20 m, N 12 m, W 6 m, S - = 23 m, E - = 14 m, N 23 m, E 14 m, N The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST R q
- 34. •Example A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 1 . 28 ) 5333 . 0 ( 5333 . 0 15 8 / 17 15 8 1 2 2 Tan Tan s m Rv 15 m/s, N 8.0 m/s, W Rv q The Final Answer : 17 m/s, @ 28.1 degrees West of North
- 35. •Example A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. S s m C V opp E s m C H adj hyp opp hyp adj hypotenuse side opposite hypotenuse side adjacent , / 64 . 33 32 sin 5 . 63 . . , / 85 . 53 32 cos 5 . 63 . . sin cos sine cosine 63.5 m/s 32˚ H.C. =? V.C. = ?
- 36. •Example A storm system moves 5000 km due east, then shifts course at 40 degrees North of East for 1500 km. Calculate the storm's resultant displacement. N km C V opp E km C H adj hyp opp hyp adj hypotenuse side opposite hypotenuse side adjacent , 2 . 964 40 sin 1500 . . , 1 . 1149 40 cos 1500 . . sin cos sine cosine R 6149.12 964.22 6224.2km Tan 964.2 6149.1 0.157 Tan 1 (0.157) 8.92o 5000 km, E 40 1500 km H.C. V.C. 5000 km + 1149.1 km = 6149.1 km 6149.1 km 964.2 km R q The Final Answer: 6224.2 km @ 8.92 degrees, North of East
- 37. What do you notice? 𝒂 2𝒂 𝒂+2 𝒃 2𝒂+4𝒃 𝒂 − 3 2 𝒂 ! Vectors are parallel if one is a multiple of another. ?
- 38. Calculating Vectors Graphically (Triangle Method- Head to Tail) For two given vectors, the resultant vector can be calculated using a scale drawing. Step 1: Choose a scale which fits the page Step 2: For example, use 1 cm = 1 m or 1 cm = 1 N Step 3: Link the vectors head-to-tail if they aren’t already Step 4: Draw the resultant vector by joining the tail of the first vector to the head of the second vector. Step 5: Measure the length of the resultant vector using a ruler. Step 6: Measure the angle of the resultant vector using a protractor Step 7: The final answer is always converted back to the units needed in the diagram. Eg. For a scale of 1 cm = 2 km, a resultant vector with a length of 5 cm measured on your ruler is actually 10 km in the scenario Example: Find the resultant of vectors a and b below
- 39. Combining Vectors by Calculation In this method, a diagram is still essential, but it does not need to be exactly to scale. The diagram can take the form of a sketch, as long as the resultant, component and sides are clearly labelled Use Pythagoras' Theorem to find the resultant vector Use trigonometry to find the angle
- 40. Step 1: Draw a vector diagram Step 2: Calculate the magnitude of the resultant vector using Pythagoras’ theorem Step 3: Calculate the direction of the resultant vector using trigonometry Step 4: State the final answer complete with direction Resultant vector = 12 km 59° east and upwards from the horizontal