Biology – Premed Windsor University School of Medicine and Health Sciences Jamaine Rowe Course Instructor.

Pre Med – Physics Chapter 1 Kinematics There is more to lectures than the power point slides! Engage your mind

Describing motion

Motion-The Particle Model A simplifying model in which we treat the object as if all its mass were concentrated at a single point. This model helps us concentrate on the overall motion of the object. Slide 1-16

Position and Time The position of an object is located along a coordinate system. At each time t, the object is at some particular position. We are free to choose the origin of time (i.e., when t = 0). Slide 1-17

Displacement The change in the position of an object as it moves from initial position x i to final position x f is its displacement ∆x = x f – x i. Slide 1-18

Checking Understanding Maria is at position x = 23 m. She then undergoes a displacement ∆x = – 50 m. What is her final position? A.–27 m B.–50 m C.23 m D.73 m Slide 1-19 Answer: Maria is at position x = 23 m. She then undergoes a displacement ∆x = – 50 m. What is her final position? A.–27 m B.–50 m C.23 m D.73 m

Speed of a Moving Object The car moves 40 m in 1 s. Its speed is = m 1 s m s The bike moves 20 m in 1 s. Its speed is = m 1 s m s

Velocity of a Moving Object Slide 1-26

Vectors A quantity that requires both a magnitude (or size) and a direction can be represented by a vector. Graphically, we represent a vector by an arrow. The velocity of this car is 100 m/s (magnitude) to the left (direction). This boy pushes on his friend with a force of 25 N to the right. Slide 1-32

Displacement is a vector Velocity is a vector

Displacement Vectors A displacement vector starts at an object’s initial position and ends at its final position. It doesn’t matter what the object did in between these two positions. In motion diagrams, the displacement vectors span successive particle positions. Slide 1-33

Vectors versus scalars : A vector is a quantity that has both a magnitude and a direction. Velocity is an example of a vector quantity. Force is a vector A scalar is just a number (no direction). The mass of an object is an example of a scalar quantity. Volume is a scalar

Vectors To graphically represent a vector, draw a directed line segment. The length of the line can be used to represent the vector’s length or magnitude.

Notation: Vector: The magnitude of a vector: Scalar: m (not bold face; no arrow) The direction of vector might be “35  south of east”; “20  above the +x-axis”; or….

Adding vectors To add vectors graphically they must be placed “tip to tail”. The result (F 1 + F 2 ) points from the tail of the first vector to the tip of the second vector. For collinear vectors: F1F1 F net ? F2F2 F1F1 F2F2

Adding Vectors Length a Length b a 2 + b 2 = c 2 Tan θ =b/a Θ b a c

Adding Vectors  Can also add Same result as before Same result as before

Components a θ b c a=c cosθ b=c sinθ a 2 +b 2 = c 2

Displacement and Distance  Displacement is the vector that points from a body’s initial position, x 0, to its final position, x. The length of the displacement vector is equal to the shortest distance between the two positions.  x = x –x 0 Note: The length of  x is (in general) not the same as distance traveled !

Average Speed and Velocity  Average speed is a measure of how fast an object moves on average: average speed = distance/elapsed time Average speed does not take into account the direction of motion from the initial and final position.

Average Speed and Velocity  Average velocity describes how the displacement of an object changes over time: average velocity = displacement/elapsed time v av = (x-x 0 ) / (t-t 0 ) =  x /  t Average velocity also takes into account the direction of motion. Note: The magnitude of v av is (in general) not the same as the average speed !

Instantaneous Velocity and Speed  Average velocity and speed do not convey any information about how fast the object moves at a specific point in time.  The velocity at an instant can be obtained from the average velocity by considering smaller and smaller time intervals, i.e. Instantaneous velocity: v = lim  t-> 0  x /  t Instantaneous speed is the magnitude of v.

Concept Question  If the average velocity of a car during a trip along a straight road is positive, is it possible for the instantaneous velocity at some time during the trip to be negative? 1 - Yes 2 - No If the driver has to put the car in reverse and back up some time during the trip, then the car has a negative velocity. However, since the car travels a distance from home in a certain amount of time, the average velocity will be positive. correct

Acceleration  Average acceleration describes how the velocity of an object moving from the initial position to the final position changes on average over time: a av = (v-v 0 ) / (t-t 0 ) =  v /  t  The acceleration at an instant can be obtained from the average acceleration by considering smaller and smaller time intervals, i.e. Instantaneous acceleration: a = lim  t-> 0  v /  t

Concept Question  If the velocity of some object is not zero, can its acceleration ever be zero ? 1 - Yes 2 - No correct If the object is moving at a constant velocity, then the acceleration is zero.

Concept Question  Is it possible for an object to have a positive velocity at the same time as it has a negative acceleration? 1 - Yes 2 – No correct An object, like a car, can be moving forward giving it a positive velocity, but then brake, causing deccelaration which is negative.

Kinematics in One Dimension Constant Acceleration  Simplifications: In one dimension all vectors in the previous equations can be replaced by their scalar component along one axis. For motion with constant acceleration, average and instantaneous acceleration are equal. For motion with constant acceleration, the rate with which velocity changes is constant, i.e. does not change over time. The average velocity is then simply given as v av = (v 0 +v)/2

Kinematics in One Dimension Constant Acceleration Consider an object which moves from the initial position x 0, at time t 0 with velocity v 0, with constant acceleration along a straight line. How does displacement and velocity of this object change with time ? a = (v-v 0 ) / (t-t 0 ) => v(t) = v 0 + a (t-t 0 ) (1) v av = (x-x 0 ) / (t-t 0 ) = (v+v 0 )/2 => x(t) = x 0 + (t-t 0 ) (v+v 0 )/2 (2) Use Eq. (1) to replace v in Eq.(2): x(t) = x 0 + (t-t 0 ) v 0 + a/2 (t-t 0 ) 2 (3) Use Eq. (1) to replace (t-t 0 ) in Eq.(2): v 2 = v a (x-x 0 ) (4)

Summary of Concepts  kinematics: A description of motion  position: your coordinates  displacement:  x = change of position  velocity: rate of change of position  average :  x/  t  instantaneous: slope of x vs. t  acceleration: rate of change of velocity  average:  v/  t  instantaneous: slope of v vs. t

Vectors Vectors are graphically represented by arrows:  The direction of the physical quantity is given by the direction of the arrow.  The magnitude of the quantity is given by the length of the arrow.

Graphical Method - Example You are told to walk due east for 50 paces, then 30 degrees north of east for 38 paces, and then due south for 30 paces. What is the magnitude and direction of your total displacement ?

Addition of Vectors  Using components (A,B lie in x,y plane): C = A+B = A x + A y + B x + B y = C x +C y Cx and Cy are called vector components of C. They are two perpendicular vectors that are parallel to the x and y axis. A x,A y and B x, B y are vector components of A and B.

Scalar Components of a Vector (in 2 dim.)  Scalar components of vector A: A = A x x +A y y | A|,  known: |A x |= |A| Cos  |A y |=|A| Sin  A x, A y known: A 2 =(A x ) 2 +(A Y ) 2  = Tan -1 |A y |/|A x |

Addition of Vectors  Using scalar components (A,B lie in x,y plane): C = A+B = A x x + A y y+ B x x+ B y y= C x x+C y y 1. Determine scalar components of A and B. 2. Calculate scalar components of C : C x = A x +B x and C y =A y +B y 3. Calculate |C| and  : C 2 =(C x ) 2 +(C Y ) 2  = Tan -1 |C y |/|C x |

Displacement and Distance  Displacement is the vector that points from a body’s initial position to its final position. The length of is equal to the shortest distance between the two positions.  x = x –x 0 The length of  x is not the same as distance traveled !

Average Speed and Velocity  Average velocity describes how the displacement of an object changes over time:  average velocity = displacement/elapsed time v = (x-x 0 ) / (t-t 0 ) =  x /  t  Average velocity also takes into account the direction of motion.  The magnitude of v is not the same as the average speed !

Summary of Concepts  kinematics: A description of motion  position: your coordinates  displacement:  x = change of position  velocity: rate of change of position  average :  x/  t  instantaneous: slope of x vs. t  acceleration: rate of change of velocity  average:  v/  t  instantaneous: slope of v vs. t