MECHANICS.

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MECHANICS

Motion Along a Straight Line
Ps 41

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 has 4 significant figures Ex. 2,500 has 2 significant figures Ex 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

Example: Young and Freedman Problem 1.31/1.38

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

Example

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

Cross Product by Components

Determinant form

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

Motion Along a Straight Line
Ps 41

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  km/h (Sport version. Picture only shows regular version)

x-t Graph Average Velocity Instantaneous 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 t t1 t

Sample Problem A Bugatti Veyron 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

Motion at Constant Acceleration

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 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

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 6th ed)
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

Chapter 2 Problem 31 A runner hopes to complete a 10,000m run in less than min. After exactly 27.0 min, there are still 1100m to go. The runner must then accelerate at 0.20m/s2 for how many seconds in order to achieve the desired time?

Chapter 2 Problem 31 Find average v at 27 min
The runner will then accelerate for t seconds covering some distance d, and will then cover the remaining distance in (180-t). (a is now zero)

Chapter 2 Problem 31

Chapter 2 Problem 31 Almost there Quadratic Equation
Results in t=357s or t=3.11s t<180 seconds so t=3.11s whew

Chapter 2 Problem 44 A falling stone takes 0.28s to travel past a window 2.2m tall. From what height above the top of the window did the stone fall?

Chapter 2 Problem 44 Let tw be the time it takes for the stone to reach the top of the window. xw be the height above the window stone was dropped At t=tw+0.28s the stone is now xw-2.2m!!!

Chapter 2 Problem 44