PHY 151: Lecture 2A Kinematics 2.1 Position, Velocity, and Speed 2.2 Instantaneous Velocity and Speed 2.3 Particle under Constant Velocity 2.4 Acceleration

Why Study Motion? Everything in the universe is moving To understand the universe, we must understand motion Historically, motion was the first classical physics topic that was understood

Mechanics Branch of physics that deals with motion  Kinematics Description of motion  Dynamics What causes motion or changes motion

Kinematics Describes motion while ignoring the external agents that might have caused or modified the motion For now, will consider motion in one dimension –Along a straight line Motion represents a continual change in an object’s position

Types of Motion Translational –An example is a car traveling on a highway Rotational –An example is the Earth’s spin on its axis Vibrational –An example is the back-and-forth movement of a pendulum

Representations of the Motion of Car Various representations include: –Pictorial –Graphical –Tablular –Mathematical The goal in many problems

Alternative Representations Using alternative representations is often an excellent strategy for understanding a problem –For example, the car problem uses multiple representations Pictorial representation Graphical representation Tabular representation Goal is often a mathematical representation

Particle We will treat a moving object as a particle –A particle is a point-like object; has mass but infinitesimal size

PHY 151: Lecture 2A Motion in One Dimension 2.1 Position, Velocity, and Speed

Kinematics Description of Motion To describe motion we will define:  Position  Distance and Displacement  Change of Distance and Displacement over time Speed and Velocity  Change of Velocity over time Acceleration

Position The object’s position is its location with respect to a chosen reference point –Consider the point to be the origin of a coordinate system Only interested in the car’s translational motion

Initial Position Type of Quantity: Vector Initial position of an object is indicted by a position vector, x i, from the origin to the position x i Magnitude of x i is distance from the origin to position x i Direction of x i is either + or - Symbol: x i SI Unit: meter (m) Note: Subscript f means the final position, x f xixi xixi

Position-Time Graph The position-time graph shows the motion of the particle (car) The smooth curve is a guess as to what happened between the data points

Data Table The table gives the actual data collected during the motion of the object (car) Positive is defined as being to the right

Distance Type of Quantity: Scalar Total path length traversed in moving from one position to another Distance depends on path and not on starting and ending points Symbol: d SI Unit: meter (m)

Distance - Example What is the distance for each of the following trips?  1.3 m east, 4 m east  2.3 m east, 4 m west For each trip, the distance is 7 m

Displacement Displacement is defined as the change in position during some time interval Type of Quantity: Vector Vector drawn from initial position to final position  x i +  x = x f   x = x f - x i Magnitude equals shortest distance between initial and final positions Direction points from initial to final position Displacement depends on the initial and final positions and not on path length Symbol:  x SI Unit: meter (m)   is always final value minus initial value and can be positive or negative XfXf XiXi XX

Direction of Displacement Vector nature of displacement is giving by  + sign to indicate displacement in the +x direction, to the right, or east  - sign to indicate displacement in the –x direction, to the left, or west Note: The same is true for position, velocity, and acceleration

Distance vs. Displacement – An Example Assume a player moves from one end of the court to the other and back Distance is twice the length of the court –Distance is always positive Displacement is zero –Δx = x f – x i =0 since x f =x i

Distance vs. Displacement An object can move so that the path length (distance) is large but the displacement is zero or small

Displacement – Example 1 What is the displacement for each of the following situations?  1.3 m east, 4 m east »Displacement is 7 m east  2.3 m east, 4 m west »Displacement is 1 m west

Displacement – Example 2 A student throws a rock straight upward from shoulder level, which is 1.65 m above the ground What is the displacement of the rock when it hits the ground?  Displacement is a vector that points from the initial position to the final position  Initial position is 1.65 m above the ground  Final position is the ground, height 0 m  Magnitude of the displacement vector is then 1.65 m  Direction of the displacement vector is downward

Average Speed Type of Quantity: Scalar Average Speed = Total Distance / Time s avg = d /  t Symbol: s avg SI Unit: meter/s (m/s) Average speed has no direction and is always positive Average speed gives details about the trip described

Average Speed – Example 1 A race car circles 10 times around an 8- km track in 1200 s What is its average speed in m/s?  distance traveled is d = 10 x 8 km = 80 km = m   t = 1200 s  s = d/  t = m / 1200 s = 66.7 m/s

Average Speed – Example 2 A motorist drives 150 km from one city to another in 2.5 h, but makes the return trip in only 2.0 h What are the average speeds for  (a)each half of the round-trip, and  (b)the total trip? (a) First half of trip  average speed = d/  t = 150 km/2.5 h = 60 km/h  Second half of trip  average speed = d/  t = 150 km/2 h = 75 km/h (b) Entire trip  average speed is d/  t = ( ) / ( ) = km/h

Average Velocity The average velocity is the rate at which displacement changes This also the slope of the line in a position-time graph Type of Quantity: Vector Average Velocity = Displacement / Time v avg  x/  t = (x f – x i )/  t Direction is the same as the direction of displacement Symbol: v avg SI Unit: meter/second (m/s)

Average Velocity – Example 1 A race car circles 10 times around an 8- km track in 1200 s What is its average velocity in m/s  After 10 laps, the ending position is the same as the starting position  Therefore, displacement is zero, even though distance is m  Since displacement is zero, average velocity is zero

Average Velocity – Example 2 Car travels half a lap in 3 seconds along a circle with radius = 150 m (a)What is the distance traveled by the car? Distance is the path length = half the circumference of the circle d = 2  r/2 =  (150) = 471 m (b)What is the magnitude of the car’s displacement? Displacement is the vector from the initial position to the final position Vector is across the diameter of the circle Magnitude of displacement is 2 x 150 = 300 m (c)What is the average speed of the car? Average speed s = distance / time = m / 3 s = 157 m/s (d)What is the magnitude of the average velocity of the car? Average velocity = displacement / time = 300 m / 3 s = 100 m/s

Average Velocity – Example 3 A car travels a full lap in 5 seconds along a circle that has a radius of 150 m? (a)What is the distance traveled by the car? Distance is the path length = circumference of the circle d = 2  r = 2  (150) = 942 m (b)What is the magnitude of the car’s displacement? Displacement is vector from the initial position to the final position The initial and final positions are at the same location Magnitude of displacement vector is zero. (c)What is the average speed of the car? Average speed s = distance / time = m / 5 s = 189 m/s (d)What is magnitude of the average velocity of car? Average velocity = displacement / time = 0 m / 5 s = 0 m/s

Average Speed and Average Velocity The average speed is not the magnitude of the average velocity. –For example, a runner ends at her starting point. –Her displacement is zero. –Therefore, her velocity is zero. –However, the distance traveled is not zero, so the speed is not zero.

PHY 151: Lecture 2A Motion in One Dimension 2.2 Instantaneous Velocity and Speed

Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time

Instantaneous Velocity, graph The instantaneous velocity is the slope of the line tangent to the x vs. t curve –This would be the green line The light blue lines show that as  t gets smaller, they approach the green line

A Note About Slopes The slope of a graph of physical data represents the ratio of change in the quantity represented on the vertical axis to the change in the quantity represented by the horizontal axis The slope has units –Unless both axes have the same units

Instantaneous Velocity, equations The general equation for instantaneous velocity is: The instantaneous velocity can be positive, negative, or zero

Instantaneous Speed The instantaneous speed is the magnitude of the instantaneous velocity The instantaneous speed has no direction associated with it

Vocabulary Note “Velocity” and “speed” will indicate instantaneous values Average will be used when the average velocity or average speed is indicated

PHY 151: Lecture 2A Motion in One Dimension 2.3 Particle under Constant Velocity

Particle Under Constant Velocity Constant velocity indicates the instantaneous velocity at any instant during a time interval is the same as the average velocity during that time interval –v x = v xavg –The mathematical representation of this situation is the equation –Common practice is to let t i = 0 and the equation becomes: x f = x i + v x t (for constant v x )

Particle Under Constant Velocity, Graph The graph represents the motion of a particle under constant velocity The slope of the graph is the value of the constant velocity The y-intercept is x i

Particle Under Constant Speed A particle under constant velocity moves with a constant speed along a straight line A particle can also move with a constant speed along a curved path The primary equation is the same as for average speed, with the average speed replaced by the constant speed

PHY 151: Lecture 2A Motion in One Dimension 2.4 Acceleration

Definition of Average Acceleration Acceleration is the rate of change of the velocity In one dimension, positive and negative can be used to indicate direction Type of Quantity: Vector Acceleration = Change in Velocity / Time a xavg  v x /  t = (v xf – v xi )/(t f – t i ) Symbol: a avg SI Unit: meter/second 2 (m/s 2 ) Note: When velocity and acceleration have opposite direction, the object is slowing

Instantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as  t approaches 0 The term acceleration will mean instantaneous acceleration –If average acceleration is wanted, the word average will be included

Instantaneous Acceleration – graph The slope of the velocity-time graph is the acceleration The green line represents the instantaneous acceleration The blue line is the average acceleration

Graphical Comparison Given the displacement- time graph (a) The velocity-time graph is found by measuring the slope of the position-time graph at every instant The acceleration-time graph is found by measuring the slope of the velocity-time graph at every instant

Acceleration and Velocity, Directions When an object’s velocity and acceleration are in the same direction, the object is speeding up When an object’s velocity and acceleration are in the opposite direction, the object is slowing down

Notes About Acceleration Negative acceleration does not necessarily mean the object is slowing down –If the acceleration and velocity are both negative, the object is speeding up The word deceleration has the connotation of slowing down –This word will not be used in the text