Or What’s Our Vector Victor? The Vector Lecture Or What’s Our Vector Victor? 1b Students know that when forces are balanced, no acceleration occurs, thus an object continues to move at a constant speed or stays at rest. 1i* Students know how to solve 2 dimensional projectile problems 1j* Students know how to resolve two dimensional vectors into their components and calculate the magnitude and direction of a vector from its components.

Vectors vs. Scalar Quantities Vector quantities have both a magnitude (measure) and direction. Vector quantities add geometrically Can be broken into components E.G. displacement, velocity, acceleration, force Scalar quantities have magnitude only Scalar quantities add algebraically (simply add values) Can have positive and negative values. E.G. mass, time Question: What distinguishes a quantity as a scalar or a vector? How do scalar quantities add? How do vector quantities add? Which type of quantity can be broken up into components (part that is in a particular direction)? Give examples of vector and scalar quantities. Activities: 1) State that vector quantities have direction and magnitude while scalar quantities only have magnitude. Discuss how magnitude is a measurement, a number, or a strength. 2) Go over the test where adding direction words such as left, right, north, south, up and down to the quantity should make sense if it is a vector quantity. Do the examples of velocity, displacement, acceleration, time and mass using this rule. 3) Discuss how because they have direction, vector quantities add geometrically while scalar quantities add algebraically. Give example of quantities of 3 and 4. If scalar, only answer is 7. If vectors answers could be -1, 1, 5 or 7 or any number between -1 and 7. 4) Discuss how vectors can be broken into horizontal or vertical components and this is often useful like for projectile motion and when dealing with forces. 5) State that scalars can have negative and positive values since the zero position is arbitrary.

Check Question 1 Which of the pictured vectors are in the same direction? A and B B and C A and C Which of the pictured vectors are equal in magnitude? A B C

Vector Addition For vectors in same or opposite direction simply add or subtract. For vectors at right angles use pythagorus to add and get resultant. For vectors at angles other than right angle break up vector(s) into components, add and then use pythagorus. Graphically can always use tip to tail and using scaling for magnitude. Questions: How do you add vector quantities that are: a) in the same direction b) in opposite directions c) at right angles to one another? Can you add a velocity to an acceleration vector? Activites: 1) Discuss how vectors are represented with an arrow where the direction of the arrow indicates the direction of the quantity and the length of the arrow indicates the magnitude of the quantity. Draw three vectors as shown in check question and ask which have the same direction and which have the same magnitude. Discuss that can move arrow and as long as direction and length are the same it is the same vector quantity. 2) When adding vectors the first rule is that your vectors MUST be the same type. Add velocity to velocity, displacement to displacement NEVER acceleration to velocity. 3) Go through airplane scenario with first a tailwind, then a headwind and finally a cross wind. Draw individual vectors and then add tip to tail to get final values. Sum up by saying when vectors are in same direction add them, when in opposite direction subtract them and when at right angles use pythagorus. 4) Finish by drawing two vectors with a wind that is partially a tail wind and partially a cross wind. State that can solve graphically using tip to tail rule but would like a mathematical way to find the magnitude. Will need to come back to this one after go over the opposite of vector addition, vector decomposition.

Check Question 2 Which of the following represents the resulting vector for the two vectors to the right? C B A

Vector Decomposition Dividing one vector into two component vectors Components are amount of vector quantity in that direction Perpendicular components do not affect one another Parallel Component = R cosine angle Perpendicular Component = R sine angle Opposite of vector addition R sine angle R Angle Question: What does it mean to decompose a vector? What does each component vector represent? What is so special about perpendicular component vectors? What are the formulas used to find perpendicular component vectors? Activities: Remind students that vector quantities have some part of them going in one direction and another part of them going in another direction. If we take one vector and make it into two vectors in those directions we have decomposed the vector. The two smaller vectors are called component vectors. Draw a displacement vector to the northeast and ask how far the hiker traveled to the east and to the north. Draw a coordinate plane and remind students of the sine and cosine functions they’ve used to find the two sides of a triangle given its hypotenuse. We can use the same equations to find the east and north components of this displacement. If necessary, draw a right triangle and derive the sine and cosine ratios. Another way to look at components is that they are the shadow of the resultant vector on the axes. If useful repeat the above with the weight of a mass on an inclined plane. 4) Draw a vector and show that there are many combinations of components that will give the resultant. State that we often choose to draw perpendicular components because they are independent of one another meaning changing one will not affect the other. We’ll demonstrate this property for motion vectors tomorrow. 5) Point out that decomposition of a vector is the opposite process to the addition of vectors. Here we are trying to find two vectors that when added will give the same effect to a body as the one resultant vector. R cosine angle

Vector Addition Revisited To add vectors not in same direction or at right angles. Decompose each vector into perpendicular components Add components in same direction Find resultant using Pythagorean theorem Find angle using tangent or other trig function and components This is where the law of sines and cosines came from Px Py Qx P Q This slide for Physics E only. Question: How can components be used to mathematically add vector quantities which are not parallel or at right angles? Write the final formula for doing this calculation of magnitude. How is the angle of the resulting vector found? Activities: Remind students of airplane velocity problem we could not solve mathematically Draw two vectors, then draw components. Draw resulting vector and show that sum of vertical components make up adjacent side of right triangle and sum of horizontal components make up opposite side of right triangle Write distance equation with opposite and adjacent sides, then substitute components and finally take square root of both side. Qy

Use of Velocity Vectors: Wind and Wave Power Fastest point of sail is nearly perpendicular to wind Running with wind means you can only go as fast as the wind Running sideways to the wind means that wind continues to push on you even when you are going the same speed as wind A B Question: In what direction is the fastest point of sail for a sail boat? Activities: Explain question and then show video (7 minutes) Explain that moving sideways to the wind means that there is still a force on the sail even when the boat is moving the same speed or faster than the wind. Same is true for surfing or Boogie Boarding a wave. Go across the wave for fastest speeds. Relate Boogie board story if have time. Ask question which of the above boats will move the fastest? C