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SI units and their prefixes. SI units and their prefixes. Vectors and scalars. Displacement and velocity. A runner completes one lap of an athletics track. What distance has she run?. 400 m. What is her final displacement?.
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Displacement and velocity A runner completes one lap of an athletics track. What distance has she run? 400m 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?
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 anddirections. 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 1N = 1kgms-2.
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.
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: y The value of a vector can be written in magnitude and direction form: 6 e.g. v = (v, θ) a Or as a pair of values called components: x 8 8 e.g. a = (8, 6) or 6
Vector addition Displacement vectors can always be added ‘nose to tail’ to find a total or resultantvector. 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: 2 -2 0 c + d = + = 3 2 5
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 3km and a horizontal displacement of 4km? magnitude: direction: R2 = 32 + 42 tan θ = 4/3 R 4km θ = tan-1(4/3) R = √ 32 + 42 = √ 25 = 53° θ = 5km 3km
vertical component horizontal component 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.
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 6ms-1 and a direction of 60° from the horizontal? cos60° = x / 6 x = 6 × cos60 sin60° = y / 6 y = 6 × sin60 6ms-1 y = 3ms-1 = 5.2ms-1 60° x
