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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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- Spring '11