Physics 101 Lecture 02A

Physics of Motion Mechanics

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 Effect that forces have on motion

1 Dimension Kinematics

Kinematics Description of Motion To describe motion we will use:  Words  Formulas  Charts  Graphs

Kinematics Description of Motion by Words and Formulas

Kinematics Description of Motion Words/Formulas 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

Kinematics Definition of 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)

Kinematics 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 north  3.3 m east, 4 m west For each trip, the distance is 7 m

1-Dimensional Kinematics Definition of 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 Xi xixi xixi

2-Dimensional Example Position / Free Vectors

1-Dimensional Kinematics Definition of Displacement 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 XfXf XiXi XX

Distance / Displacement

1-Dimension Kinematics 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

1-Dimension Kinematics Example of + / - Sign 500 meters in the +x direction is +500 meters 500 meters in the –x direction is -500 meters

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

1-Dimension Kinematics 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 north »Displacement is 5 m at 53 0 above +x axis  3.3 m east, 4 m west »Displacement is 1 m west

1-Dimension Kinematics 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

Kinematics Definition of Average Speed Type of Quantity: Scalar Average Speed = Distance / Time s avg = d / t Symbol: s avg SI Unit: meter/s (m/s)

What Does Constant Speed Mean? Ball has initial position x i = 0 m at time t i = 0 Ball has constant velocity t = 5 m/s This means that during :  1 st second ball moves 5 m  2 nd second ball moves 5 m  3 rd second ball moves 5 m, and so on … Position at time t is:  t = 0 s, x = 0 m  t = 1 s, x = 5 m  t = 2 s, x = 10 m  t = 3 s, x = 15 m

1-Dimension Kinematics 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 = 80,000 m  t = 1200 s  s avg = d/t = / 1200 s = 66.7 m/s

1-Dimensional Kinematics 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

1-Dimensional Kinematics Definition of Average Velocity 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)

1-Dimension Kinematics 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.

1-Dimensional Kinematics 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

1-Dimension Kinematics 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

1-Dimension Kinematics Definition of Average Acceleration Type of Quantity: Vector Acceleration = Change in Velocity / Time a avg  v/t = (v f – v i )/t Symbol: a avg SI Unit: meter/second 2 (m/s 2 ) Note: When velocity and acceleration have opposite direction, the object is decelerating

1-Dimension Kinematics Average Acceleration – Example 1 An automobile traveling at 8 m/s along a straight, level road accelerates to 20 m/s in 6.00 s What is the magnitude of the auto’s average acceleration?  average acceleration =  (v f – v i )/t = (20 – 8) / 6 = 2 m/s 2

1-Dimension Kinematics Average Acceleration – Example 2 Motorcycle has a constant acceleration of 2.5 m/s 2 Both the velocity and acceleration of the motorcycle point in the same direction How much time is required for the motorcycle to change its velocity from 21 to 31 m/s.  a avg = (v f – v i ) / t  2.5 m/s 2 = (31 m/s – 21 m/s) / t  t = 10 / 2.5 = 4 s

Kinematics Description of Motion by Charts and Graphs

1-Dimension Kinematics Graphical Analysis 1 Velocity = 0Acceleration = 0 Horizontal line on P-T graph: velocity = 0 Time Pos555555

1-Dimension Kinematics Graphical Analysis 2 Velocity = 1Acceleration = 0 Sloping line on P-T graph: velocity <> 0 Time Pos012345

1-Dimension Kinematics Graphical Analysis 3 Velocity = 2Acceleration = 0 Steeper slope means greater velocity Time Pos

1-Dimension Kinematics Graphical Analysis 4 Velocity = -1 Acceleration = 0 Negative slope means velocity in opposite direction Time Pos

1-Dimension Kinematics Graphical Analysis 5 Velocity = 1 Acceleration = 1 Curving line on P-T graph: acceleration <> 0 Time Pos

1-Dimension Kinematics Speed vs Average Speed (Charts) Time (min) Position (m)Time (min) Speed (m/min)

1-Dimension Kinematics Speed vs. Avg. Speed (Graphs)

1-Dimension Kinematics Speed vs. Average Speed Prior charts & graphs show trip of car During trip the velocity changes Average velocity remains constant during the trip Average velocity produces same final position as does the varying velocity