lecture3_9_13_11 - Lecture 3 note text in red refers to the...

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Unformatted text preview: Lecture 3: 09/13/11 note: text in red refers to the textbook, 9th edition extended. 1. Obtaining position from velocity – section 2-10 Starting with the definition of velocity () ( ) () () rearrange to get: ( ) () () ie, the increment in the position between time t and t+dt is given by the product of the velocity at t and the time interval dt. This is true for small intervals of time dt. We can therefore get the position at a time t2 relative to an initial position t1 by breaking the time interval into little bits of time and adding ( ) for each bit of time. () () ∑ () ∫ () (definition of the integral). Graphically, this is the area under the v-t curve between t1 and t2 2. Integral as an antiderivative - section 2-8 Given the velocity as a function of time ( ), the position as a function of time ( ) is () ∫ () where c is an arbitrary constant. The integral of a function can therefore be defined as an antiderivative – the integral of a function results in a new function, such that when I take the derivative of this new function I end up with the function I am integrating. Common integrals we will use are: ∫ ∫ () () ∫ Each of these can be checked by taking the derivative of both sides: if ∫ ( ) () () . ( ), then 3. Two dimensional vectors – section 3-2 4. Components of vectors - section 3-4 Given a vector ⃗ that has components and (written as | ⃗ | or simply ) is given by and the angle it makes with the x axis along the x and y axes, the magnitude of the vector is given by () Equivalently we can get the components and from the magnitude and direction as () () 5. Adding vectors – section 3-3, 3-6 Given two vectors ⃗ and ⃗⃗, the sum of the two vectors ⃗ component sums: ⃗⃗ is given simply in terms of the ⃗ The magnitude of is ( We define the quantity ( ) ( ) ( ) ) as the “dot product” of the two vectors ⃗ and ⃗⃗ ⃗ ⃗⃗ ( See section 3-7 and 3-8 for properties of dot products. ) The dot product is a convenient mathematical tool to express many vector equations. ...
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