Name: _____________________________________________ Period: _______ Table #:__________ Unit 1b: Kinematics Equations © 2006 Bill Amend Unit 1b Schedule – Kinematics Equations * If you are absent, your teacher expects you to check the calendar, go on-line, get the notes and attempt the classwork* BE PROACTIVE, NOT REACTIVE!! 9/13 Friday Unit 1a Test-Intro and Graph Shapes Week 1 Date Day In Class Activity Homework Monday 9/17 Tuesday Activity 1b--Practice with Graphical Analysis-Part 1 Start Activity 1b -Part 2 Activity 1b--Practice with Graphical Analysis-Part 2 Start Activity #2b Ex 2b 9/16 9/18 9/19 Wednesday Falcon Time Thursday (Pep Assembly) EDP: Kine Video #6-Linearizing Ex 1b Ex 3b Activity #2b Transfer Data to GA Work on modified report for Activity 2 Work on modified report for Activity 2b EDP: Kine Video #7-Deriving Kinematics Equations 9/20 Friday Institute Day Week 2 Date Day In Class Activity 9/23 Monday 9/24 Tuesday 9/25 Wednesday Falcon Time 9/26 Thursday 9/27 Friday Review Kinematic Equations Introduce 4th equation Sample Problems Go over Ex# 4b Discuss Vertical Motion Ex #5b Go over Ex# 5b Exercise #6b Go over Ex #6b Exercise #7b Motion Test Homework Ex #4b EDP: Kine Video #8-Vertical Motion Finish Ex# 4b Google Quiz: Vertical Motion Finish Ex# 5b Finish Ex #6b-Check answers online Study EDP: PJM Video #1 Curve Fitting Reference Sheet When checking a law or determining a functional relationship, there is good reason to believe that a uniform curve or straight line will result. The process of matching an equation to a curve is called curve fitting. The desired empirical formula, assuming good data, can usually be determined by inspection. There are other mathematical methods of curve fitting, however they are very complex and will not be considered here. Curve fitting by inspection requires an assumption that the curve represents a linear or simple power function. If data plotted on rectangular coordinates yields a straight line, the function y = f(x) is said to be linear and the line on the graph could be represented algebraically by the slope-intercept form: y = mx + b, where m is the slope and b is y-intercept. Consider the following graph of velocity vs. time: 10 vel (m/s) 5 0 5 time (s) 10 The curve is a straight line, indicating that v = f(t) is a linear relationship. Therefore, v = mt + b, where slope = m = Ошибка!= Ошибка! From the graph, m = Ошибка!= 0.80 m/sОшибка!. The curve intercepts the v-axis at v = 2.0 m/s. This indicates that the velocity was 2.0 m/s when the first measurement was taken; that is, when t = 0. Thus, b = v0 = 2.0 m/s. The equation of the line, v = mt + b, can then be rewritten as the mathematical equation: v = (0.80 m/s2 )t + 2.0 m/s. The general equation would be: V = at + vo Identifying a variable from its units Unit m m/s Kg N Nm Ns Quantity Displacement Velocity Mass Force Energy Impulse (Change in Momentum) Variable x v m F J ∆p Unit m/s/s m/s2 kgm/s2 N/kg N/m kgm/s Quantity Acceleration Acceleration Force Grav. Field Const. Spring Constant Momentum Variable a a F J k p When trying to identify what the slope or y-intercept means or represents, use the chart above. Find the unit and match the term and variable accordingly so you can write the general equation and determine the meaning of the slope, yintercept, area under the curve and relationship between your variables. Example: Based on the graph, the units for the slope are N/m. If you look at the chart that and find the units, the slope represents the spring constant (k). Also, based on the graph, the units for the Yintercept are N so they represent a force (F). As a result you would get a general equation that looks like: F = kx + F0 Exercise #1b: Straightening out a Graph The chart below identifies the different types of graph shapes you may encounter when entering your data into Graphical Analysis. The main purpose of most of the labs we do, is to be able to recognize what graph shape you have created with your data and how to linearize it. Use your notes and fill in this chart below. Graph Shape and Name Straight line Top opening parabola Side opening parabola Hyperbola Relationship between the variables How to linearize the graph The Equation of the Line Activity 1b--Graphing Data-Part 1 1. Graph the original data. Assume the 1st column is the independent variable and the 2nd column the dependent variable. 2. Determine from the graph shape the calculation you will use to linearize your data. Record those calculated values in the manipulated variables column. 3. Graph the new calculated data 3. Write the mathematical equation for each data set. Be sure to have the linear fit box open to get your values for slope and y-intercept. Use the template for data set 1 and 2 as your guide to set up your mathematical equation. 4. Then using the mathematical equation, do the calculation asked. Show ALL work! Sketch the graph below. Data Set 1: Displacement and Force Displacement (m) Force (N) 0.05 0.98 0.15 2.94 0.18 3.53 0.22 4.31 0.25 4.9 Mathematical expression Data set 1 y = m (# and unit) x = Calculate the displacement when the force is 3.5 N. + + b (# and unit) Data set 2: Time and Position Manipulated Variable Time (s) Position (m) 1 6.25 2 15 3 26.25 4 40 5 56.25 Original Graph Linear Graph Mathematical expression Data set 2 y = m (# and unit) x = Calculate the Position when the velocity is 15 m/s. + + b (# and unit) Data set 3: Time and Velocity Manipulated Variable time (sec) velocity (m/s) .3 10 1.2 20 2.7 30 4.8 40 7.5 50 Original Graph Linear Graph Mathematical expression Data set 3 y = m (# and unit) x = Calculate the Velocity when the time is 9 seconds. + + b (# and unit) Data set 4: Mass and Acceleration Manipulated Variable mass (Kg) acceleration (m/s/s) 0.1 40 0.5 8 1 4 2 2 4 1 Original Graph Linear Graph Mathematical expression Data Set 4 y = m (# and unit) x = Calculate the acceleration when the mass is 6 Kg. + + b (# and unit) Exercise #2b: Straightening out a Graph-Without GA 1. The mass values of specified volumes of pure gold nuggets are given in the table below. Volume (cm3) 1.0 2.0 3.0 4.0 5.0 Mass (g) 19.4 38.6 58.1 77.4 96.5 A. Plot mass (y- axis) v. Volume (x-axis) from the values given in the table and draw the curve that best fits the point. B. Describe the resulting graph shape. (Use Ex 1) C. According to the graph, what type of relationship (direct, inverse, exponential?) exists between the mass and the volume of the pure gold nuggets? D. Write the mathematical equation for the LINEAR graph. y = m (# and unit) = x + b (# and unit) + F. Write a word interpretation for the slope of the line specific to this graph information. G. What is the volume of 200g of the above substance? 2. In a biology experiment, the number of yeast cells is determined after 24 hours of culture at different temperatures. At each temperature change, the # of yeast cells were counted and recorded below. Amount of yeast (# of cells) Temperature (0C) 25 0 35 10 45 20 55 30 A. What is the independent variable? (x-axis) & what is the dependent variable? (y-axis) B. Create a graph of the data above. Label the variables on your x and y axes. C. Calculate the slope of your line; be sure to include units. D. What is the meaning of the slope? E. What is the value (#) and unit of the y-intercept? F. What is the meaning of the y-intercept? (Your answer should not be where the line crosses the y-axis!) G. Write the mathematical equation for the line. Y = = m (# and unit) x + b (# and unit) + H. Mathematically determine how many yeast cells would be present, after 24 hours, if the culture temperature were 38oC Activity 1b--Graphing Data-Part 2 1. Graph the original data. Assume the 1st column is the independent variable and the 2nd column the dependent variable. 2. Manipulate your graph to straighten out the curve (use your Reference Packet directions) 3. Write the mathematical equation for each data set. Be sure to have the linear fit box open to get your values for slope and y-intercept. Use the template for data set 1 and 2 as your guide to set up your mathematical equation. Data set 1 Volume and Pressure V (m3) P (pa) Data set 2 Time and Position Manipulated Variable t (s) x (m) .1 40 .1 .03 .5 8 .2 .12 1 4 .5 .75 2 2 1 3 4 1 2 12 5 .8 3 27 8 .5 4 48 10 .4 5 75 Manipulated Variable Mathematical expression Data Set 1 y = m (# and unit) X = + b (# and unit) + Mathematical expression Data set 2 y = = m (# and unit) X + + b (# and unit) Data set 3 Age and Weight A (months) W (lbs) Manipulated Variable Data set 4 Time and Velocity t (s) v (m/s) 1 7 .3 10 2 9.4 1.2 20 3 10.5 2.7 30 4 12.0 4.8 40 5 13.0 7.5 50 6 14.3 10.8 60 7 15.2 14.7 70 8 16.7 19.2 80 Mathematical expression #3 Mathematical expression #4 Manipulated Variable Exercise #3b: Equation Writing Practice 1. Velocity vs Time 2. Acceleration vs 1/Mass Equation: _________________________________ Equation: _______________________________ Original relationship: _______________________ Original relationship: _____________________ 3. Displacement vs Time2 4. Velocity2 vs Position Equation: _________________________________ Equation: _______________________________ Original relationship: _______________________ Original relationship: _____________________ Activity #2b: Equation Lab Purpose: To Derive the Kinematic Equations. Things to think about while doing this lab: Work quickly and efficiently, do not waste time What should be the acceleration of a dropping ball? Basic Procedure Open Pasco Capstone and choose “Two Small and One Large Display”. Click on “Hardware Setup”, click on port 1, and choose “Photogate”. Click on “Timer Setup” and click “Next” for the first 2 options. o For option 3, choose “Picket Fence” o For option 4, ONLY “Speed” and “Position” should be checked o For option 5, set “Flag Spacing” to 0.05 m and click “Next” o For option6, click “Finish” and then click “Timer Setup” again to close the screen On your display, have a velocity/time graph on your screen, along with tables for velocity and position. o Click the icon on the large display and choose “Graph”. Make the y-axis “Speed” and the x-axis should automatically turn into time. o Make the small displays into tables for Speed/Time, and Position/Time Click “Record” and drop a Picket Fence through the photogate. Your data should automatically appear. o You will know your data is good if your graph is linear. Copy and paste the correct section of data into Graphical Analysis. In Graphical Analysis, make graphs of: o Position vs. Corrected Time o Velocity vs. Corrected Time o Velocity vs. Position (Put velocity on the y-axis) ***You will need to make a calculated “corrected time” column by subtracting your first time value from all of the rest of the time values!! Straighten out those graphs and determine the equation for the line of each graph Looking at your units and values and knowing what the acceleration of the object should be, determine the general equations. Sketch all initial and straightened graphs in your notebook with appropriate quantitative data. Exercise #4b: Horizontal Motion Problems 1. A car accelerates from 13 m/s to 25 m/s in 6.0 s. a. What was its acceleration? b. How far did it travel in this time? 2. A car slows down from 23 m/s to rest in a distance of 85 m. What was its acceleration? 3. A small plane must reach a speed of 33 m/s for takeoff. What is the minimum length the runway can be if the acceleration is 3.0 m/s2? 4. A world-class sprinter can burst out of the blocks to essentially top speed (11.5 m/s) in the first 15.0 m of the race. a. What is the acceleration of the sprinter? b. How long does it take her to reach that speed? 5. A car slows down uniformly from a speed of 21.0 m/s to rest in 6.0 s. How far did it travel in that time? 6. In coming to a stop, a car leaves skid marks 92 m long on the highway. If it slowed at a rate of 7.0 m/s2, what was the initial velocity of the car? 7. A car traveling 85 km/hr strikes a tree. The front end of the car compresses and the driver comes to rest after traveling 0.80 m. What was the average acceleration of the driver during the collision? 8. Determine the stopping distances for a car with an initial speed of 95 km/hr and human reaction time of 1.0 s for an acceleration of (a) -4.0 m/s2 and (b) -8.0 m/s2. (Hint: this is a 2-part problem…what happens during the reaction time?) Exercise #5b: Vertical Motion Problems 1. A stone is dropped from the top of a cliff. It hits the ground below after 3.25 s. How high is the cliff? 2. If a car rolls gently (Vo = 0 m/s) off a vertical cliff, how long does it take it to reach 85 km/hr. 3. King Kong falls straight down from the top of the Empire State Building (380 m high). a. How long did it take King Kong to fall? b. What was his velocity just before landing? 4. A baseball is hit straight up into the air with a speed of 22 m/s. a. How high does it go? b. How long does it take before the ball returns to its starting position? 5. A ballplayer catches (at the same height) a ball 3.0 s after throwing it upward. a. With what speed did he throw the ball? b. What maximum height did it reach? 6. An object starts from rest and falls under the influence of gravity. Draw graphs of (a) its velocity and (b) the distance it has fallen from 0-5 seconds. 7. A helicopter is ascending vertically with a speed of 5.2 m/s. At a height of 125 m above the earth, a package is dropped from a window. How much time does it take for the package to reach the ground? (Hint: Package and helicopter have the same initial speed). Exercise #6b: Kinematics Practice 1. A person driving her car at 45 km/hr approaches an intersection just as the traffic light turns yellow. She knows that the yellow light only lasts 2.0 s before turning red, and she is 28 m away from the near side of the intersection. Should she try to stop, or should she speed up to cross the intersection before the light turns red? The intersection is 15 m wide. Her car’s maximum acceleration is -5.8 m/s2 when slowing down, and can accelerate from 45 km/hr to 65 km/hr in 6 s. (Ignore the length of her car and reaction time.) 2. 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 11.0 m? b. How long does it take to reach this height? Why are there 2 answers? 3. A falling stone takes 0.28 s to travel past a window 2.2 m tall. From what height above the top of the window did the stone fall? (Hint: Break this into 2 parts: passing the window and falling from the top.) 4. Suppose you point the nozzle vertically upward at a height of 1.5 m above the ground. When you move the nozzle away from the vertical, you hear the water striking the ground for another 2.0 s. What is the water speed as it leaves the nozzle? 5. A stone is thrown vertically upward with a speed of 12.0 m/s from the edge of a cliff 70.0 m high. a. What is the total distance it travels? b. What is its speed just before hitting the ground? c. How much later does it reach the bottom of the cliff? 6. A baseball passes a window 28 m above street level going upward at 13 m/s. If the ball was thrown from the street: a. What was its initial speed? b. What maximum height does it reach? c. When does it hit the street again after passing the window? Exercise #7b: Kinematics Review Problems 1. A moving car begins to brake at a constant rate with an acceleration of -4.6m/s2 and slows to a stop in a distance of 78m. a. What was the initial speed of the car? b. How much time in seconds did it take the car to stop? c. If the car’s initial velocity was instead 31m/s, but the rate of acceleration was the same as above, how far would the car travel before coming to a stop? d. With the data from part ( c ), would the car be able to stop in time for a squirrel crossing the road 100m away from the point where the car starts braking? If so, how far from Mr. Squirrel would the car be when it stopped? If not, what rate of acceleration would be needed in order to save Mr. Squirrel? 2. A water balloon is shot vertically upward from a cliff that is 45m high. The water balloon’s initial velocity is 8.6m/s. a. What is the maximum height that the water balloon reaches with respect to its launching point? b. How much time in seconds would it take for the water balloon to get to the maximum height? c. How much time in seconds would it take for the water balloon to hit the ground? (Total time from when it was first shot.) d. What total distance does the water balloon travel? e. What is the velocity of the balloon when it hits the ground at the bottom of the cliff? 3. Little Johnny Throckmorton starts off from rest on his bike, and accelerates toward his friend’s house for 10 seconds at a rate of 1.0m/s2. After 10 seconds, he stops accelerating and travels at a constant speed for another 100 seconds before reaching his friend’s house. After staying there for only one minute, Johnny and his friend continued on in the same direction to a second friend’s house with an average velocity of 7m/s. This trip took 3 minutes. a. What was Johnny’s maximum velocity? b. How far away from Johnny’s house was the first friend’s house? c. What is the distance between the first and second friends’ houses? d. What is Johnny’s total displacement? e. What Johnny’s average velocity for the entire trip? 4. A ball is thrown vertically upward with a velocity of 15 m/s off a cliff and lands at the bottom 6 seconds later. a. What is the ball’s maximum height with respect to its launching point? b. What is the height of the cliff? c. What is the ball’s velocity when it hits at the bottom of the cliff? d. After how many seconds will the ball again be at the same height as where it was thrown from? 5. Given the position vs. time graph… a. Calculate the velocity of each segment of the graph (from 0 s to 10 s, from 10 s to 15 s, etc.) b. Draw a corresponding graph representing velocity vs. time. c. What is the acceleration value for this graph? How do you know? d. What is the total distance covered? e. What is the overall displacement of the object? 6. A stone is thrown vertically downward with a speed of 15 m/s from a cliff that is 100 m high. a. What is the stone’s acceleration? b. What is the velocity of the stone just before it reaches the ground? c. How much time, in seconds, does it take the stone to reach the ground?