Lecture2 - General Physics(PHY 2130 Lecture II Lecture II...

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Unformatted text preview: General Physics (PHY 2130) Lecture II Lecture II • Math review (vectors cont.) • Motion in one dimension Position and displacement Velocity average instantaneous Acceleration motion with constant acceleration • Motion in two dimensions Lightning Review Lightning Review Last lecture: 1. Math review: trigonometry 2. Math review: vectors, vector addition Note: magnitudes do not add unless vectors point in the same direction 1. Physics introduction: units, measurements, etc. someone’s pulse for 10 sec instead of a minute? Review Problem: How many beats would you detect if you take Hint: Normal heartbeat rate is 60 beats/minute. Components of a Vector Components of a Vector ► A component is a part ► It is useful to use rectangular components ► Vector A is now a sum of its These are the projections of the vector along the x­ and y­axes components: A = Ax + Ay Ax Ay What are and ? The components are the legs of the right triangle whose hypotenuse is A Ay 2 2 −1 A = A x + A y and θ = tan Ax ► The x­component of a vector is the projection along the x­axis ► Components of a Vector Components of a Vector Ax = A cosθ The y­component of a vector is the projection along the y­axis ► Ay Ay = A sin θ ► Then, A = Ax + Ay Notes About Components Notes About Components ► The previous equations are valid only if θ is measured with respect to the x­axis ► The components can be positive or negative and will have the same units as the original vector Example 1 A golfer takes two putts to get his ball into the hole once he is on the green. The first putt displaces the ball 6.00 m east, and the second, 5.40 m south. What displacement would have been needed to get the ball into the hole on the first putt? Given: ∆ x1= 6.00 m (east) ∆ x2= 5.40 m (south) Solution: 1. Note right triangle, use Pythagorean theorem 6.00 m 5.40 m R = ( 6.00 m ) + ( 5.40 m ) = 8.07 m 2 2 Find: R = ? 2. Find angle: 5.40 m −1 ° θ = tan −1 = tan ( 0.900) = 42.0 6.00 m What Components Are Good For: What Components Are Good For: Adding Vectors Algebraically ► Choose a coordinate system and sketch the ► Find the x­ and y­components of all the vectors ► Add all the x­components vectors v1, v2, … This gives Rx: Rx = ∑ v x Ry = ∑ v y ► Add all the y­components Magnitudes of vectors pointing in the same direction can be added to find the resultant! This gives Ry: Adding Vectors Algebraically (cont.) Adding Vectors Algebraically (cont.) ► Use the Pythagorean Theorem to find the magnitude of the Resultant: R = R2 + R2 x y ► Use the inverse tangent function to find the direction of R: θ = tan −1 Ry Rx IV. Motion in One Dimension Dynamics Dynamics ► The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts In kinematics, you are interested in the description of motion Not concerned with the cause of the motion ► Kinematics is a part of dynamics Position and Displacement Position and Displacement ► Position is defined in terms A of a frame of reference Frame A: x >0 and x >0 B y’ i O’ f x’ Position and Displacement Position and Displacement ► Position is defined in terms of a frame of reference One dimensional, so generally the x­ or y­axis ► Displacement measures the change in position Represented as ∆ x (if horizontal) or ∆ y (if vertical) Vector quantity (i.e. needs directional information) indicate direction for one­ dimensional motion Units SI CGS US Cust Meters (m) Centimeters (cm) Feet (ft) ► + or ­ is generally sufficient to Displacement Displacement measures Displacement represented as ∆ x or ∆ y the change in position ∆x1 = x f − xi = 80 m − 10 m = + 70 m ∆x2 = x f − xi = 20 m − 80 m = − 60 m Distance or Displacement? Distance or Displacement? ► Distance may be, but is not necessarily, the magnitude of the displacement Displacement (yellow line) Distance (blue line) Position­time graphs Position­time graphs Note: position-time graph is not necessarily a straight line, even though the motion is along x-direction ConcepTest 1 ConcepTest 1 An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is 1. 2. 3. 4. 5. either greater than or equal to always greater than always equal to either smaller or equal to either smaller or larger than the distance it traveled. Please fill your answer as question 1 of General Purpose Answer Sheet ConcepTest 1 ConcepTest 1 An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is 1. 2. 3. 4. 5. either greater than or equal to always greater than always equal to either smaller or equal to either smaller or larger than the distance it traveled. Please fill your answer as question 2 of General Purpose Answer Sheet ConcepTest 1 (answer) ConcepTest 1 (answer) An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is 1. 2. 3. 4. 5. either greater than or equal to always greater than always equal to either smaller or equal to either smaller or larger than the distance it traveled. Note: displacement is a vector from the final to initial points, distance is total path traversed Average Velocity Average Velocity ► It takes time for an object to undergo a displacement ► The average velocity is rate at which the displacement occurs vaverage ∆x x f − xi = = ∆t ∆t ► Direction will be the same as the direction of the displacement (∆ t is always positive) More About Average Velocity More About Average Velocity ► Units of velocity: Units SI CGS US Customary Meters per second (m/s) Centimeters per second (cm/s) Feet per second (ft/s) ► Note: other units may be given in a problem, but generally will need to be converted to these Example: Example: Suppose that in both cases truck covers the distance in 10 seconds: ∆x1 + 70m v1 average = = ∆t 10s = +7m s ∆x2 − 60m v2 average = = ∆t 10 s = −6m s Speed Speed ► Speed is a scalar quantity (no information about sign/direction is need) ► Speed is the magnitude of the velocity same units as velocity Average speed = total distance / total time Graphical Interpretation of Average Velocity Graphical Interpretation of Average Velocity ► Velocity can be determined from a position­ time graph vaverage ∆x + 40m = = ∆t 3 .0 s = + 13 m s ► Average velocity equals the slope of the line joining the initial and final positions Instantaneous Velocity Instantaneous Velocity ► Instantaneous velocity is defined as the limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero vinst x f − xi ∆x = lim = lim ∆t →0 ∆t ∆t →0 ∆t ► The instantaneous velocity indicates what is happening at every point of time Uniform Velocity Uniform Velocity ► Uniformvelocity is constant velocity ► The instantaneous velocities are always the same All the instantaneous velocities will also equal the average velocity Graphical Interpretation of Instantaneous Graphical Interpretation of Instantaneous Velocity ► Instantaneous velocity is the slope of the tangent to the curve at the time of interest ► The instantaneous speed is the magnitude of the instantaneous velocity Average vs Instantaneous Velocity Average vs Instantaneous Velocity Average velocity Instantaneous velocity Let’s watch the movie! Let’s watch the movie! ConcepTest 2 ConcepTest 2 The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: 1. 2. 3. 4. 5. at time tB both trains have the same velocity both trains speed up all the time both trains have the same velocity at some time before tB train A is longer than train B all of the above statements are true A B position tB time Please fill your answer as question 3 of General Purpose Answer Sheet ConcepTest 2 ConcepTest 2 The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: 1. 2. 3. 4. 5. at time tB both trains have the same velocity both trains speed up all the time both trains have the same velocity at some time before tB train A is longer than train B all of the above statements are true A B position tB time Please fill your answer as question 4 of General Purpose Answer Sheet ConcepTest 2 (answer) ConcepTest 2 (answer) The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: 1. 2. 3. 4. 5. at time tB both trains have the same velocity both trains speed up all the time both trains have the same velocity at some time before tB train A is longer than train B all of the above statements are true A B position Note: the slope of curve B is parallel to line A at some point t< tB tB time Average Acceleration Average Acceleration ► Changing velocity (non­uniform) means an acceleration is present ► Average acceleration is the rate of change of the velocity aaverage ∆v v f − vi = = ∆t ∆t ► Average acceleration is a vector quantity (i.e. described by both magnitude and direction) Average Acceleration Average Acceleration ► When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing ► When the sign of the velocity and the acceleration are opposite, the speed is decreasing Units SI CGS US Customary Meters per second squared (m/s2) Centimeters per second squared (cm/s2) Feet per second squared (ft/s2) Instantaneous and Uniform Instantaneous and Uniform Acceleration ► Instantaneous acceleration is the limit of the average acceleration as the time interval goes to zero ainst v f − vi ∆v = lim = lim ∆t →0 ∆t ∆t →0 ∆t ► When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average acceleration Graphical Interpretation of Graphical Interpretation of Acceleration ► Average acceleration is the slope of the line connecting the initial and final velocities on a velocity­time graph ► Instantaneous acceleration is the slope of the tangent to the curve of the velocity­time graph Example 1: Motion Diagrams Example 1: Motion Diagrams ► Uniform velocity (shown by red arrows maintaining the same size) ► Acceleration equals zero Example 2: Example 2: ► ► ► Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Example 3: Example 3: ► ► ► Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Let’s watch the movie! Let’s watch the movie! One­dimensional Motion With Constant One­dimensional Motion With Constant Acceleration ► a=a If acceleration is uniform (i.e. ): v f − vo f 0 a= = thus: t −t t v f − vo v f = vo + at Shows velocity as a function of acceleration and time One­dimensional Motion With Constant One­dimensional Motion With Constant Acceleration ► Used in situations with uniform acceleration v f = vo + at vo + v f ∆x = vaveraget = 2 t Velocity changes uniformly!!! 12 ∆x = vot + at 2 2 v 2 = vo + 2a∆x f Notes on the equations Notes on the equations ∆x = vaverage ► Gives displacement as a function of velocity and time vo + v f t = t 2 12 ∆x = vot + at 2 ► Gives displacement as a function of time, velocity and acceleration 2 v 2 = vo + 2a∆x f ► Gives velocity as a function of acceleration and displacement Summary of kinematic equations Summary of kinematic equations Free Fall 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 ► This acceleration is called the acceleration due to gravity, and indicated by g Acceleration due to Gravity Acceleration due to Gravity ► Symbolized by g ► g = 9.8 m/s² (can use g = 10 m/s² for estimates) ► g is always directed downward toward the center of the earth Free Fall ­­ an Object Dropped Free Fall ­­ an Object Dropped ► Initial velocity is zero y ► Frame: let up be positive ► Use the kinematic equations Generally use y instead of x since vertical x vo= 0 a=g 12 ∆y = at 2 a = −9.8 m s 2 Free Fall ­­ an Object Thrown Free Fall ­­ an Object Thrown Downward ► a = g With upward being positive, acceleration will be negative, g = ­9.8 m/s² With upward being positive, initial velocity will be negative ► Initial velocity ≠ 0 Free Fall ­­ object thrown upward Free Fall ­­ object thrown upward ► Initial velocity is upward, so positive ► The instantaneous velocity at the maximum height is zero ► a = g everywhere in the motion v=0 g is always downward, negative Thrown upward Thrown upward ► The motion may be symmetrical then tup = tdown then vf = ­vo ► The motion may not be symmetrical Break the motion into various parts ►generally up and down Non­symmetrical Non­symmetrical Free Fall ► Need to divide the motion into segments ► Possibilities include Upward and downward portions The symmetrical portion back to the release point and then the non­ symmetrical portion Combination Motions Combination Motions ConcepTest 3 ConcepTest 3 A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit ground below the cliff with greater speed is the one initially thrown 1. upward 2. downward 3. neither – they both hit at the same speed Please fill your answer as question 5 of General Purpose Answer Sheet ConcepTest 3 ConcepTest 3 A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit ground below the cliff with greater speed is the one initially thrown 1. upward 2. downward 3. neither – they both hit at the same speed Please fill your answer as question 6 of General Purpose Answer Sheet ConcepTest 3 (answer) ConcepTest 3 (answer) A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit ground below the cliff with greater speed is the one initially thrown 1. upward 2. downward 3. neither – they both hit at the same speed Note: upon the descent, the velocity of an object thrown straight up with an initial velocity v is exactly –v when it passes the point at which it was first released. ConcepTest 3 (answer) ConcepTest 3 (answer) A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit ground below the cliff with greater speed is the one initially thrown 1. upward 2. downward 3. neither – they both hit at the same speed Note: upon the descent, the velocity of an object thrown straight up with an initial velocity v is exactly –v when it passes the point at which it was first released. Fun QuickLab: Reaction time Fun QuickLab: Reaction time 12 d = g t , g = 9.8 m s 2 2 2d t= g ...
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This note was uploaded on 04/12/2011 for the course PHYS 2130 taught by Professor Petrov during the Spring '11 term at Wayne State University.

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