9/2/2009USF Physics 100 Lecture 31 Physics 100 Fall 2009 Lecture 3 Kinematics in One Dimension Gravity and Freely Falling Bodies

9/2/2009USF Physics 100 Lecture 32 Thanks for the cartoon to Moose’s, 1652 Stockton St., San Francisco, CA

9/2/2009USF Physics 100 Lecture 33 Agenda (1) Administrative Prof. Benton is in Germany Substitute Lecturer this week: Terrence A. Mulera – HR 102 Office Hours: Today 1-2 PM and by appointment Contact Information:   Phone: (415) st Lab week of 14 September Homework ? Syllabus is on wiki

9/2/2009USF Physics 100 Lecture 34 Agenda (2) Today’s Lecture  Kinematics in One Dimension Define Kinematics Displacement Velocity and/or Speed Acceleration Gravity and Falling Bodies  Tools for More Than One Dimension Trigonometry Review, Vectors Dave’s Short Trig Course,  Movie

9/2/2009USF Physics 100 Lecture 35 Kinematics Kinematics  Description of motion with no reference to forces Dynamics  Effect of forces on motion Kinematics + Dynamics → Mechanics

9/2/2009USF Physics 100 Lecture 36 Displacement Vector pointing from an object’s initial position to its final position In 1-D, say x x0x0 x xx Magnitude |  x |, is shortest distance from x 0 to x or vice-versa. Scalar Only thing vectorial in 1-D is ±. Also true for velocity and acceleration in 1-D. Units: Length, e.g. m, cm, km, ft, in, miles, etc.

9/2/2009USF Physics 100 Lecture 37 Velocity Time rate of change of displacement Vector Note that velocity is an instantaneous quantity and can itself vary with time (Calculus fans only) Digression on derivatives and tangent lines

9/2/2009USF Physics 100 Lecture 38 Average velocity The magnitude of velocity is called speed. It is a scalar which reveals nothing about the direction of motion Typical units of both velocity and speed are m/sec, ft/sec, etc.

9/2/2009USF Physics 100 Lecture 39 Acceleration Time rate of change of velocity Vector Note that acceleration is an instantaneous quantity and can itself vary with time (Calculus fans only) Time rate of change of acceleration is called jerk Time rate of change of jerk is called surge

9/2/2009USF Physics 100 Lecture 310 Constant Acceleration t vfvf v0v0 slope = a v For constant acceleration,  v above is really the average value

9/2/2009USF Physics 100 Lecture 311

9/2/2009USF Physics 100 Lecture 312 Summary of Units Displacement length m Velocity length / time m / sec Acceleration length / time 2 m / sec 2 Quantity Type of Unit Example Dimensional Analysis: Units in an equation expressing a physical process must “make sense”. e.gmakes sense makes no sense

9/2/2009USF Physics 100 Lecture 313

9/2/2009USF Physics 100 Lecture 314 Gravitational Acceleration We are getting a bit ahead of ourselves here talking about forces Near the surface of the Earth For falling distances small w.r.t. the Earth’s radius, r E Acceleration due to gravity, (-)  downward

9/2/2009USF Physics 100 Lecture m /sec cm / sec 2 32 ft /sec 2 { Acceleration due to gravity on Earth Different for other celestial objects depending on their mass and size Approximate values:

9/2/2009USF Physics 100 Lecture 316 Freely Falling Bodies Acceleration due to gravity: g = m/sec 2 Dropping an object from a height. After a time t (constant) g Units: (m/sec 2 ) (sec) → m/sec Velocity: Distance dropped: y = ½ g t 2 Units: (m/sec 2 ) (sec 2 ) → m

9/2/2009USF Physics 100 Lecture 317 Throw an object straight up with velocity v 0 How long does it stay up? How high does it go? Symmetry => time going up = time coming down initial velocity = - final velocity velocity at apex = 0 Rearranging and Remember that g is (-). Then the time comes out (+)

9/2/2009USF Physics 100 Lecture 318 Again, remember that g is (-). Then the height comes out (+)

9/2/2009USF Physics 100 Lecture 319 Trigonometry Review Angles:  (+)  counter-clockwise (-)  clockwise Units: 360º in a complete circle 2  radians. Arc length in terms of radius Note that  + 360º or  + 2  rad. = .  is periodic.

9/2/2009USF Physics 100 Lecture 320 Triangles: a b c B A C a + b + c = 180º or  radians Right Triangle:  Pythagorean Theorem: B 2 = A 2 + C 2 B  hypotenuse A  side opposite C  side adjacent w.r.t. 

9/2/2009USF Physics 100 Lecture 321 Trigonometric Functions: sin  = side opposite / hypotenuse cos  = side adjacent / hypotenuse tan  = side opposite / side adjacent cot  = side adjacent / side opposite sec  = hypotenuse / side adjacent csc  = hypotenuse / side opposite Note: The co-functions are the functions of the complement of . e.g. cos  = sin (90º -  ), etc.

9/2/2009USF Physics 100 Lecture 322 sin t cos t Periodic Phase difference of 90° 0r  /2

9/2/2009USF Physics 100 Lecture 323 Applications: Measurement of Height:  h d Measure d and , then h = d tan 

9/2/2009USF Physics 100 Lecture 324 Vectors Scalars: Magnitude only e. g. mass, energy, temperature Vectors: Magnitude and direction e. g. force, momentum Graphically: Length represents magnitude Orientation represents direction 

9/2/2009USF Physics 100 Lecture 325 Graphical Addition of Vectors A BC The parallelogram of forces

9/2/2009USF Physics 100 Lecture 326 Component Addition of Vectors x-axis y-axis AxAx AyAy AA A x = A cos  A A y = A sin  A Similarly, B x = B cos  B, etc.

9/2/2009USF Physics 100 Lecture 327 Add the components C x = A x + B x C y = A y + B y

9/2/2009USF Physics 100 Lecture 328 Unit Vector Notation Cartesian coordinates: corresponding to Unit vectors. i.e. z y x

9/2/2009USF Physics 100 Lecture 329 Multiplication of Vectors Scalar( a ) times Vector( A ): Vector( ) times Vector( ) → Scalar( S ) Scalar or dot product Note that the magnitude squared of a vector is simply  Scalar

9/2/2009USF Physics 100 Lecture 330 Vector( ) times Vector( ) → Vector( ) Vector or cross product Note that this yields another vector (actually an axial vector)

9/2/2009USF Physics 100 Lecture 331 A C  To the plane of AB Right Hand Rule Direction of advance of a right hand screw

9/2/2009USF Physics 100 Lecture 332 Note that the vector product is not commutative Again look at Right Hand Rule ‛ A‛ A C A × B B × A