Transcript Vectors
1 of 24 © Boardworks Ltd 2009 2 of 24 © Boardworks Ltd 2009 SI units and their prefixes 3 of 24 © Boardworks Ltd 2009 SI units and their prefixes 4 of 24 © Boardworks Ltd 2009 Vectors and scalars 5 of 24 © Boardworks Ltd 2009 Displacement and velocity A runner completes one lap of an athletics track. What distance has she run? 400 m What is her final displacement? If she ends up exactly where she started, her displacement from her starting position is zero. What is her average velocity for the lap, and how does it compare to her average speed? 6 of 24 © Boardworks Ltd 2009 Vector equations An equation is a statement of complete equality. The left hand side must match the right hand side in both quantity and units. In a vector equation, the vectors on both sides of the equation must have equal magnitudes and directions. Take Newton’s second law, for example: force = mass × acceleration Force and acceleration are both vectors, so their directions will be equal. Mass is a scalar: it scales the right-hand side of the equation so that both quantities are equal. Force is measured in newtons (N), mass in kilograms (kg), and acceleration in ms-2. The units on both sides must be equal, so 1 N = 1 kgms-2. 7 of 24 © Boardworks Ltd 2009 Scalar or vector? 8 of 24 © Boardworks Ltd 2009 9 of 24 © Boardworks Ltd 2009 Displacement vectors Harry and Sally are exploring the desert. They need to reach an oasis, but choose to take different routes. Harry travels due north, then due east. N Sally simply travels in a straight line to the oasis. When Harry met Sally at the oasis, they had travelled different distances. However, because they both reached the same destination from the same starting point, their overall displacements were the same. 10 of 24 © Boardworks Ltd 2009 Vector notation A scalar quantity is often represented by a lower case letter, e.g. speed, v. A vector quantity can also be represented by a lower case letter, but it is written or printed in one of the following formats to differentiate it from the scalar equivalent: The value of a vector can be written in magnitude and direction form: e.g. v = (v, θ) 11 of 24 6 a Or as a pair of values called components: e.g. a = (8, 6) or y 8 6 8 x © Boardworks Ltd 2009 Vector addition Displacement vectors can always be added ‘nose to tail’ to find a total or resultant vector. y b This can be done approximately by scale drawing: a 10 a+b x 7 It can also be done by calculation, breaking each vector down into perpendicular components first and then adding these together to find the components of the resultant: c+d = 12 of 24 2 3 + -2 2 = 0 5 © Boardworks Ltd 2009 Calculating a resultant When adding two perpendicular vectors, it is often necessary to calculate the exact magnitude and direction of the resultant vector. This requires the use of Pythagoras’ theorem, and trigonometry. For example, what is the resultant vector of a vertical displacement of 3 km and a horizontal displacement of 4 km? R 4 km θ magnitude: direction: R2 = 32 + 42 tan θ = 4/3 R = √ 32 + 42 = √ 25 θ = tan-1(4/3) = 53° = 5 km 3 km 13 of 24 © Boardworks Ltd 2009 Resultant vectors 14 of 24 © Boardworks Ltd 2009 15 of 24 © Boardworks Ltd 2009 Vector components Just as it is possible to add two vectors together to get a resultant vector, it is very often useful to break a ‘diagonal’ vector into its perpendicular components. This makes it easier to describe the motion of an object, and to do any relevant calculations. 16 of 24 vertical component horizontal component © Boardworks Ltd 2009 Calculating components A vector can be separated into perpendicular components given only its magnitude and its angle from one of the component axes. This requires the use of trigonometry. For example, what are the horizontal and vertical components of a vector with a magnitude of 6 ms-1 and a direction of 60° from the horizontal? cos60° = x / 6 sin60° = y / 6 x = 6 × cos60 y = 6 × sin60 = 3 ms-1 6 ms-1 y = 5.2 ms-1 60° x 17 of 24 © Boardworks Ltd 2009 Vector components 18 of 24 © Boardworks Ltd 2009 Velocity components 19 of 24 © Boardworks Ltd 2009 Understanding vectors 20 of 24 © Boardworks Ltd 2009 21 of 24 © Boardworks Ltd 2009 Glossary 22 of 24 © Boardworks Ltd 2009 What’s the keyword? 23 of 24 © Boardworks Ltd 2009 Multiple-choice quiz 24 of 24 © Boardworks Ltd 2009