But First a Review • Significant Figures • Non-zero digits are always significant. • Any zeros between two significant digits are significant. • A final zero or trailing zeros in the decimal portion are significant. • Ex. 0.002500 has 4 significant figures • Ex. 2,500 has 2 significant figures • Ex. 2.500 x 103 has 4 significant figures • Multiplication/Division – Determined by the LEAST number of significant figures • Addition/Subtraction – Determined by LEAST number of decimal of places in the decimal portion
Vectors • Vectors are physical quantities with both magnitude and direction and cannot be represented by just a single number • Displacement vs. Distance • Velocity vs. Speed • Represented by A • The magnitude of A is represented by |A| or A P2 A P1 P2 B=-A P1
Vector Addition • Tip to tail method or Parallelogram method • Vector addition is commutative (a) (b)
Vector Components • Vector Components
Example: Young and Freedman Problem 1.31/1.38 • A postal employee drives a delivery truck along the route shown. Determine the magnitude and direction of the resultant displacement.
Example: Young and Freedman Problem 1.31/1.38 • DON’T FORGET DIRECTION
Unit Vectors • Unit vectors are unitless vectors with a magnitude of 1. • Primarily used to point a direction. • Represented by • Note scalar times vector
Dot Product • Or scalar product • Using components
Example • What is the angle between the vectors Compute up to 3 sig figs. • Solution
Cross Product • Or vector product • Direction is dictated by the right hand rule • Anti-commutative
Example • Vector A has a magnitude to 5 and lies in the direction of the x-axis. Vector B has a magnitude of 2 and lies along the xy-plane at a 30o angle with the x-axis. Find AxB. • Solution Let
Displacement • Is a vector quantity, usually denoted by x. • Change in the position of a point. (we can approximate objects to be a particle) • Remember, since it’s a vector, it’s important to note both magnitude and direction. • Define positive displacement to be a movement along the positive x-axis
Average Velocity P1 • At time t1 the car is at point P1 and at time t2 the car is at point P2 • We can define P1 and P2 to have coordinates x1 and x2 respectively Δx=x2-x1 • Average velocity P2
Velocity • Velocity is the change in displacement per unit time in a specific direction. • It is a vector quantity, usually denoted by v • Has SI unit of m/s • Average velocity can be useful but it does not paint the complete picture. • The winner of a race has the highest average velocity but is not necessarily the fastest.
Instantaneous Velocity • Velocity at a specific instant of time • Define instant as an extremely short amount of time such that it has no duration at all. • Instantaneous Velocity • top speed of 431.072 km/h (Sport version. Picture only shows regular version)
x-t Graph • Average Velocity • Average velocity is the slope of the line between two points • Instantaneous Velocity • Instantaneous velocity is the slope of the tangent line at a specific point x x x2 x1 o o t1 t2 t1 t t
Sample Problem • A BugattiVeyron is at rest 20.0m from an observer. At t=0 it begins zooming down the track in a straight line. The displacement from the observer varies according to the equation a) Find the average velocity from t=0s to t=10s b) Find the average velocity from t=5s to t=10s c) Find the instantaneous velocity at t=10s
Solution • a) • b) • c)
Acceleration • Acceleration describes the rate of change of velocity with time. • Average Acceleration • Vector quantity denoted by • Instantaneous acceleration
WARNING • Just because acceleration is positive (negative) does not mean that velocity is also positive (negative). • Just because acceleration is zero does not mean velocity is zero and vice versa.
Motion at Constant Acceleration • Assume that acceleration is constant. • Generally
Feel Free to use vf, vi, v0 whatever notation you’re more comfortable with • BUT be consistent through out the entire problem
Seat Work #1 • Using • Derive • Hint: Eliminate time
Giancoli Chapter 2 Problem 26 • In coming to a stop a car leaves skid marks 92 m long on the highway. Assuming a deceleration of 7.00m/s2, estimate the speed of the car just before braking.
Chapter 2 Problem 26 • Ignore negative
Falling Objects • Most common example of constant acceleration is free fall. • Freely falling bodies are objects moving under the influence of gravity alone. (Ignore air resistance) • Attracts everything to it at a constant rate. • Note: because it attracts objects downwards acceleration due to gravity is • Galileo Galilei formulated the laws of motion for free fall
Freely Falling • A freely falling body is any body that is being influenced by gravity alone, regardless of initial motion. • Objects thrown upward or downward or simply released are all freely falling
Example Giancoli 2-42 • A stone is thrown vertically upward with a speed of 18.0 m/s. (a) How fast is it moving when it reaches a height of 11.0m? (b) How long will it take to reach this height? (c) Why are there two answers for b?
Giancoli 2-42 • We can’t ignore negative
Giancoli 2-42 • Why were there 2 answers to b?
Summary • These 4 equations will allow you to solve any problem dealing with motion in one direction as long as acceleration is CONSTANT! • 1. • 2. • 3. • 4.
Problems from the Book (Giancoli 6thed) • 14- Calculate the average speed and average velocity of a complete round-trip, in which the outgoing 250 km is covered at 95km/hr, followed by a 1 hour lunch break and the return 250km is covered at 55km/hr. Start 95 kph 1 hour break End 55 kph
Chapter 2 Problem 14 • Average speed = change in distance / change in time • For first leg • For return • Total time • Average speed
Chapter 2 Problem 14 • What was the cars average velocity?
Chapter 2 Problem 19 • A sports car moving at constant speed travels 110m in 5.0s. If it then brakes and comes to a stop in 4.0 s, what is its acceleration in m/s2? Express the answer in terms of g’s where g=9.80 m/s2.
Chapter 2 Problem 19 • First find v