Chapter 2 Motion in One Dimension

Free Fall All objects moving under the influence of only gravity are said to be in free fall All objects moving under the influence of only gravity are said to be in free fall All objects falling near the earth’s surface fall with a constant acceleration All objects falling near the earth’s surface fall with a constant acceleration Galileo originated our present ideas about free fall from his inclined planes Galileo originated our present ideas about free fall from his inclined planes The acceleration is called the acceleration due to gravity, and indicated by g The acceleration is called the acceleration due to gravity, and indicated by g

Acceleration due to Gravity Symbolized by g Symbolized by g g = 9.8 m/s² g = 9.8 m/s² g is always directed downward g is always directed downward toward the center of the earth toward the center of the earth

Non-symmetrical Free Fall Need to divide the motion into segments Need to divide the motion into segments Possibilities include Possibilities include Upward and downward portions Upward and downward portions The symmetrical portion back to the release point and then the non- symmetrical portion The symmetrical portion back to the release point and then the non- symmetrical portion

Combination Motions

Chapter 3 Vectors and Two-Dimensional Motion

Vector Notation When handwritten, use an arrow: When handwritten, use an arrow: When printed, will be in bold print: A When printed, will be in bold print: A When dealing with just the magnitude of a vector in print, an italic letter will be used: A When dealing with just the magnitude of a vector in print, an italic letter will be used: A

Properties of Vectors Equality of Two Vectors Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected Any vector can be moved parallel to itself without being affected

Adding Vectors When adding vectors, their directions must be taken into account When adding vectors, their directions must be taken into account Units must be the same Units must be the same Graphical Methods Graphical Methods Use scale drawings Use scale drawings Algebraic Methods Algebraic Methods More convenient More convenient

Graphically Adding Vectors, cont. Continue drawing the vectors “tip-to-tail” Continue drawing the vectors “tip-to-tail” The resultant is drawn from the origin of A to the end of the last vector The resultant is drawn from the origin of A to the end of the last vector Measure the length of R and its angle Measure the length of R and its angle Use the scale factor to convert length to actual magnitude Use the scale factor to convert length to actual magnitude AF_0306.swf

Notes about Vector Addition Vectors obey the Commutative Law of Addition Vectors obey the Commutative Law of Addition The order in which the vectors are added doesn’t affect the result The order in which the vectors are added doesn’t affect the result

Vector Subtraction Special case of vector addition Special case of vector addition If A – B, then use A+(-B) If A – B, then use A+(-B) Continue with standard vector addition procedure Continue with standard vector addition procedure

Components of a Vector A component is a part A component is a part It is useful to use rectangular components It is useful to use rectangular components These are the projections of the vector along the x- and y-axes These are the projections of the vector along the x- and y-axes

Components of a Vector, cont. The x-component of a vector is the projection along the x-axis The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis The y-component of a vector is the projection along the y-axis Then, Then,

More About Components of a Vector The previous equations are valid only if θ is measured with respect to the x-axis The previous equations are valid only if θ is measured with respect to the x-axis The components can be positive or negative and will have the same units as the original vector The components can be positive or negative and will have the same units as the original vector The components are the legs of the right triangle whose hypotenuse is A The components are the legs of the right triangle whose hypotenuse is A May still have to find θ with respect to the positive x-axis May still have to find θ with respect to the positive x-axis

Adding Vectors Algebraically Grandma’s house Grandma’s house Add all the x and y-components Add all the x and y-components This gives R x and R y : This gives R x and R y : Use the Pythagorean Theorem to find the magnitude of the Resultant: Use the Pythagorean Theorem to find the magnitude of the Resultant: Use the inverse tangent function to find the direction of R: Use the inverse tangent function to find the direction of R: