Class 6: Vector and scalar, components Mapping velocity and - PowerPoint PPT Presentation

Class 6: Vector and scalar, components

Mapping velocity and acceleration on a real number line (1D) 1m/s v f v i 4m 5m -4m 2m 3m -3m -2m -1m 0 1m -5m x i = -2m x f = +4m i 4m/s 6m/s t i = 2s t f = 6s x f = -3m x i = +3m v f v i 5m 4m 3m 2m 1m -4m 0 -1m -2m -3m -5m 1m/s Velocity and acceleration can now add like real numbers on the number line. There is no confusion because we only add or subtract physical quantities of the same kind. The “coordinate” of the velocity and acceleration are now called “component”.

Extending 1D motion to 2D and 3D motions 5m 4m 3m r is the position vector, or P 2m displacement vector. r 1m -5m -4m -3m -2m -1m 1m 2m 3m 4m 5m -1m 1. Position  Displacement from Two ways to specify the position of -2m point P: origin. -3m 1. Give the coordinates ( ‐ 4m, +2m) 2. Same argument can be applied 2. Give you enough information to to velocity and acceleration. -4m draw the arrow (vector) pointing -5m from the origin to P. “Business as usual” along the x ‐ and y ‐ axis independently, but simultaneously.

Vector and scalar Physical quantities that can be represented by coordinates like position are called vectors. Physical quantities that cannot be represented by coordinates like position are called vectors. Examples of vector: Examples of scalar: Displacement Distance Velocity Speed Acceleration Mass Force Temperature Weight Energy Momentum Scalar × Vector  Vector Scalar × Scalar  Scalar Vector × Vector  Complicated

Multiplying a vector with a scalar y 6m (mr) x = m r x 3r (mr) y = m r y 2m r 1m 3m “Business as usual” along the x ‐ and y ‐ axis independently, but simultaneously. Same for other kinds of vectors, including velocity and acceleration.

Addition and subtraction of vectors Only vectors of the same kind (hence same unit) can be added or subtracted. y r 1 + r 2 (r 1 +r 2 ) x = r 1x + r 2x 12m (r 1 +r 2 ) y = r 1y + r 2y (r 1 ‐ r 2 ) x = r 1x ‐ r 2x r 1 r 1 - r 2 (r 1 ‐ r 2 ) y = r 1y ‐ r 2y 3m r 2 5m -4m “Business as usual” along the x ‐ and y ‐ axis independently, but simultaneously. Same for other kinds of vectors, including velocity and acceleration.

Right angled triangle A c b a C B       2 2 2 2 2 a b c (or sin cos 1) opposite side   sin hypotenuse adjacent side   cos hypotenuse If you know two sides, or one side and  one angle, you know everything about opposite side sin    tan the right angled triangle.  adjacent side cos     o A B 90