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Scalars vs Vectors

Scalars vs Vectors. Scalars – a quantity that only needs a magnitude (with a unit) to describe it Ex: time, distance, speed, mass, volume, temperature Vectors – a quantity that needs a magnitude (with a unit) and a direction to completely describe it Ex: displacement, velocity, acceleration

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Scalars vs Vectors

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  1. Scalars vs Vectors Scalars – a quantity that only needs a magnitude (with a unit) to describe it Ex: time, distance, speed, mass, volume, temperature Vectors – a quantity that needs a magnitude (with a unit) and a direction to completely describe it Ex: displacement, velocity, acceleration and there are others still to come… You’ve never heard of them before because Bio & Chem only deal with scalar quantities. Vectors are generic tools, developed specifically for Physics, used to help us solve problems and explain things more clearly in Physics.

  2. Vector Addition in 1D Vectors, of the same type of quantity, that are along a line, like East & West or Up & Down, can summed by simple arithmetic: Ex: 8 m/s Right + 3 m/s Right = _____________ (If Rt is chosen +, then it’s 8m/s + 3m/s = 11m/s) Ex: 5 m South + 4 m North = _____________ (If N is chosen +, then it’s -5 m + 4 m = -1 m) But vectors that aren’t along the same line, need other means by which to find their resultant…

  3. Vector Addition in 2D What’s the net or resultant displacement of someone who has walked 3 km to the East, then turned left and walked 4 km due North? Is it 7 km? No – It’s 5 km at 53˚ toward the North of due East What’s the resultant velocity of a plane that points its nose due West with an engine speed of 600 km/hr, but encounters a Southerly wind of 75 km/hr? Is it 675 km/hr? No – It’s 605 km/hr at 7˚ toward South of due West

  4. How to do Vector Addition in 2D 2 Ways to Find the resultant of vector quantities in 2D • Scaled vector diagrams • Mathematical solutions Either way you use, you need a diagram… • Tip-to-tail method – we’ll start with this • Parallelogram method – we won’t use this much • Component method – we’ll learn this soon • If doing it by SVD… then need a scaled diagram • If doing it by math, you’ll only need a sketch of the vectors, then use math like: • Pythagorean theorem: ___________________ • sin, cos, tan functions: ___________________

  5. Scaled Vector Diagrams Arrows are used to represent vector quantities in diagrams • its length represents the magnitude of the vector, according to the units set by some scale • the arrow tip indicates its direction according to some directional key Try one: Draw the velocity v = 30 km/hr, East What should we use as a scale? The scale should set up your diagram to be as large as possible in the space provided, within the constraints of the length of your c-thru. What about a directional key?

  6. N W + E S So finally, v = 30 km/hr, east would look like: scale 1 cm = 2 km/hr & key Note: the tip should be drawn very carefully so to not change the length of the carefully drawn arrow.

  7. Vector Components Consider the following: for a displacement of something that ends up SW of where it started, that could have come about by that object moving so far directly W, then so far directly S… The legs of the right triangle are the components of the original vector

  8. OR, for a plane that flies with a velocity in the NW direction, that could be because the plane’s engines where moving it W while the wind was blowing it N… Again, the legs of the right triangle are the components of the original vector

  9. Quite often in Physics it’s important to know the value of the components of a vector • there are 2 for each vector • they are vectors themselves • they’re direction is along a main direction • they’re perpendicular to each other • if added together, their sum equals the original vector Resolution - the process of determining the components of a given vector – to resolve a vector into its components Let’s try one….

  10. More Math Operators for Vectors So far we’ve concentrated on addition of 2 or more vector quantities, but there are other mathematical operations that can be performed on vectors: • Subtraction – easiest to think of it as adding the opposite – so it affects the direction, not the magnitude Ex: 3 km North – 4 km East = ? or 3 km North + (– 4 km East) = ? where the opposite of 4 km East is 4 km West so instead do 3 km North + 4 km West = ? (5 km, 53˚ W of N) Try more: What’s – 25 m/s, 40˚ up of right? What’s – 7.8 m, 23˚ right of back?

  11. Multiplication or Division of a Vector by a Scalar – affects the magnitude, but not the direction Ex: 2 (5 km/hr, 47˚ S of W) = ? 10 km/hr, 47˚ S of W Or 32 m, 19˚ down of left ÷ 4 seconds = ? 8m/s, 19˚ down of left So the answer is a vector with the same direction as the original vector, but a different magnitude You’ve actually been doing a lot of these already: v = Δx/Δt vf = vi + aΔt Δx = vi Δt + ½ a Δt2 Δx = ½ (vi + vf)/Δt

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