AP Chapter 4 Notes - Birmingham Schools
Moving on! Chapter 2Name____________________Position vs. timeVelocity vs. timeAcceleration vs. timeWhat does a Single point tell you about the motion?What does a horizontal line mean about the motion?What quantity is the Slope of the graph?What does a straight line mean about the motion?What does a curved line mean about the motion?What quantity is the area of the graph bounded by the x- axis?Know how to find average velocity. When is vi+vo2 true? Position vs timeVelocity vs timeAcceleration vs timePositionDisplacement?Average velocity?Instantaneous velocity?Average acceleration?Instantaneous acceleration?An automobile travels on a straight road for 40 km at 30 km/h. It then continues in the same direction for another 40 km at 60 km/h. What is the average velocity? Be able to calculate average velocity/acceleration, or instantaneous velocity/ acceleration given a graph of one of the kinematic quantities (position, velocity, acceleration)A particle starts from 2 m at t=0 and moves along the positive x axis. A graph of the velocity of the particle as a function of time is shown above.What is the position of the particle at t=5 s?What is the velocity of the particle at t=5 s?What is the acceleration of the particle at t=5 s?What is the average velocity of the particle between t=1 s and t= 5 s?What is the average acceleration of the particle between t=1 s and t=5 s?Given a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, recognize in what time intervals the other two are positive, negative, or zero, Sketch a graph for each of the following constant acceleration motionsGiven a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, identify or sketch a graph of each as a function of time.Focus on how the graphs are related to each other!!Position vs timeVelocity vs timeAcceleration vs timerestJustificationConstant velocityJustificationSpeeding up while traveling to the rightJustificationSpeeding up while traveling to the leftJustificationSlowing down while traveling to the rightJustificationSlowing down while traveling to the leftJustificationTurning aroundJustificationGiven a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, recognize in what time intervals the other two are positive, negative, or zero, yvo=0 xo=0Sketch a position vs. time, velocity vs. time and acceleration vs. time graph for a block sliding down a frictionless plane. xvatttvo=0 yxo=0xvatttSketch a position vs. time, velocity vs. time and acceleration vs. time graph for a block sliding down a frictionless plane.Given a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, identify or sketch a graph of each as a function of time.xo=0Sketch a position vs. time, velocity vs. time and acceleration vs. time graph for a block sliding up an incline plane then sliding back down. label each graph with motion.xvatttvo=0xo=0Sketch a position vs. time, velocity vs. time and acceleration vs. time graph for a block sliding down an inclined plane and onto a horizontal plane. label each graph with motion.xvatttGiven a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, identify or sketch a graph of each as a function of time.vo=0xo=0Sketch a position vs. time, velocity vs. time, and acceleration vs. time graph for a block sliding down an incline plane, traveling horizontally and then up the next incline plane. label each graph with motion.xvattt2472690161290The graph gives the acceleration a(t) of a Chihuahua as it chases a German shepherd along an axis. In which of the time periods indicated does the Chihuahua move at constant speed? A graph of acceleration versus time for a particle as it moves along an x axis is shown above. At t=0 the coordinate of the particle is 4m and the velocity is 2 m/s. What is the velocity of the particle at t=2s?Write the equation for the slope of the velocity vs. time graph and solve for final velocity.Write the equation for the area under the curve of a velocity time graph for an object speeding up with a constant rate. (it’s a trapezoid)What are all the constant acceleration equations?Use constant acceleration equations to solve problems509587553975The following equations describe the motion of a particle in four situations: Which of the following are constant acceleration?x=4t2+2t-6v=4t2+2t-6x=3t-4v=3t-4An object starts from rest and accelerates uniformly in the horizontal direction, moving 32 m during the first 4 s. What are the magnitudes of the object’s displacement and velocity after it accelerates for the first 3 s. Magnitude of Magnitude of DisplacementVelocity24 m 24 m/s24 m18 m/s18 m18 m/s18 m12 m/s8 m12 m/s3752850-3181354963795-455930A red train traveling at 72 km/h and a green train traveling at 144 km/h are headed toward one another along a straight, level track. When they are 950 m apart, each engineer sees the other’s train and applies the brakes. The brakes accelerate each train at the rate of -1.0 m/s2. Is there a collision? If so, what is the speed of each train at impact? If not, what is the separation between the trains when they stop?Be able to use a system of equations to solve constant acceleration problems Arlo is a sprinter with a top speed of 11 m/s. If Arlo starts from rest and accelerates at a constant rate, he is able to reach his top speed in a distance of 12 meters. He is then able to maintain this top speed for the remainder of a 100 m race. What is his time for the 100 m race?In order to improve his time, Arlo tries to decrease the distance required for him to reach his top speed. What must this distance be if he is to achieve a time of 10 s for the race?Be able to write expressions using kinematic equationsA block of mass m is projected up from the bottom of an inclined ramp with an initial velocity vo. The ramp has negligible friction and makes an angle θ with the horizontal. A motion sensor aimed down the ramp is mounted at the top of the incline so that the positive direction is down the ramp. The block starts a distance D from the motion sensor, as shown above. The block slides partway up the ramp, stops before reaching the sensor, and then slides back down.Consider the motion of the block at some time t after it has been projected up the ramp. Express your answers in terms of m, D, vo, t, θ and physical constants, as appropriate.Determine the acceleration a of the blockAcceleration = gsinθ (we will learn this later… Newton’s 2nd Law)Determine an expression for the velocity v of the blockDetermine an expression for the position x of the blockDerive an expression for the position xmin of the block when it is closest to the motion sensor. Express your answers in terms of m, D, vo, θ, and physical constants, as appropriate. Be able to use a system of equations to solve constant acceleration problems Many AP problems give you data. How do you determine what to graph? How do you draw a best fit line? Does a best fit line go through the origin? (Easy Points!!!!) Be able to draw a graph of data so that the slope has meaning, draw a best fit line and use a graph to answer questionsA nonlinear spring is compressed horizontally. The spring exerts a force that obeys the equation F = Ax1/2, where x is the distance from equilibrium that the spring is compressed and A is a constant. A physics student records data on the force exerted by the spring as it is compressed and plots the two graphs below, which include the data and the student's best-fit curves. From one or both of the given graphs, determine A. Be sure to show your work and specify the units. -483870208915Be able to draw a graph of data so that the slope has meaning, draw a best fit line and use a graph to answer questionsRm DA solid disk of unknown mass and known radius R is used as a pulley in a lab experiment, as shown above. A small block of mass m is attached to a string, the other end of which is attached to the pulley and wrapped around it several times. The block of mass m is released from rest and takes a time t to fall the distance D to the floor. Calculate the linear acceleration a of the falling block in terms of the given quantities.The time t is measured for various heights D and the data are recorded in the following table.D(m)t(s)0.50.6811.021.51.1921.38What quantities should be graphed in order to best determine the acceleration of the block? Explain your reasoning. Calculate any values not given, place them in the table and graph them below. Draw the line of best fit. -2984567310 Use your SLOPE to calculate the acceleration. Free FallWhat force acts on an object in the air? What is the direction of this force? What acceleration does this force produce? What does “2g” mean?You are throwing a ball straight up in the air. At the highest point, the ball’s a. velocity and acceleration are zerovelocity is nonzero but its acceleration is zeroacceleration is nonzero, but its velocity is zerovelocity and acceleration are both nonzero-21907549530If you drop an object in the absence of air resistance, it accelerates downward at 9.8 m/s2. If instead you throw it downward, its downward acceleration after release is less than 9.8 m/s29.8 m/s2more than 9.8 m/s2 In a classroom demonstration, a teacher has a coin and feather in a long tube. The teacher uses a vacuum pump to remove all the air from the tube. The feather and coin are then dropped from the top of the tube at the same time. Which of the following describes what happens and why?The coin hits the bottom of the tube first because it weighs moreThe coin hits the bottom of the tube first because it is more denseThe feather hits the bottom of the tube first because it weighs lessThe coin and feather hit the bottom of the tube at the same time because they weigh the sameThe coin and feather hit the bottom of the tube at the same time because they have the same accelerationAn object is dropped from rest from the top of a 400 m cliff on Earth. If air resistance is negligible what is the distance the object travels during the first 6s of its fall?What is the acceleration at any point? Where is the velocity zero? Be able to find the distance object travels.30 m60 m120 m180 m360 mIn the absence of air friction an object dropped near the surface of the Earth experiences a constant acceleration of about 9.8 m/s2. This means that thespeed of the object increases 9.8 m/s during each secondspeed of the object as it falls is 9.8 m/sobject falls 9.8 meters during each secondobject falls 9.8 meters during the first second onlyderivative of the distance with respect to time for the object equals 9.8 m/s2A startled armadillo leaps upward, rising 0.544 m in the first 0.200 s. What is its initial speed as it leaves the ground?478155080645What is its speed at the height of 0.544m?How much higher does it go?What is the acceleration at any point? Where is the velocity zero? Be able to find the distance object travels.A sphere is thrown down vertically with an initial speed of vo from a height of h.Write an expression for the velocity just before striking the ground. Would the speed just before striking the ground be greater than, less than, or the same if the initial speed was thrown upward?Would the time in the air be greater than, less than, or the same if the initial speed was thrown upward?Would the displacement of the sphere be greater than, less than, or the same if the initial speed was thrown upward? Calculus= Study of slopes (derivatives) and areas (antiderivaties/integral)What is a derivative? How do we symbolize that it is a derivative?Given the graph of the equation x=v0t+1/2 at2 find the slope between two points a time dt apart. Q xP tP(t1,x1): Q(t1+dt, x2) P(t1,v0t1+1/2 at12)Q(t1+dt, v0(t1+dt)+1/2 a (t1+dt)2 )What happens to the slope as dt goes to zero and the points merge so that you are finding the slope of a tangent line and instantaneous velocity? (You just used the definition of a derivative)Draw a graph of the derivative of x=v0t+1/2 at2. What physics quantity is the derivative/slope of this graph?What is an easy method to determine the instantaneous slope (derivative) of a graph that is respect to time?What’s the derivative of a position equation?take the derivative of x=1/2 at2+votWhat does this mean? Is this average or instantaneous? Take the derivative againWhat does this mean? Is this instantaneous or average?Given an expression for one of the kinematic quantities, position, velocity, or acceleration, as a function of time, determine the other two as a function of time (derivatives/antiderivatives), and find when these quantities achieve their maximum and minimum valuese. How do you find maximum or minimum velocity? ExampleBelow is given the equation of motion for a subatomic particle as a function of timex(t)=(20t4-10t-20)mFind the equation for the velocity of the particle. At what time is velocity zero? Given an expression for one of the kinematic quantities, position, velocity, or acceleration, as a function of time, determine the other two as a function of time (derivatives/antiderivatives), and find when these quantities are zero.Find the equation for the acceleration of the particle. At what time is acceleration zero?Find the position of the particle at 3.0 seconds.Know the difference between average and instantaneous quantities. Be able to calculate average velocity/acceleration, or instantaneous velocity/ acceleration given equations as a function of time.Determine the displacement between t=0 and t=3.00 seconds.What was the average velocity during this time interval?Find the instantaneous velocity of the particle at t=3.00 seconds. If the position of an object is given by x=2t3, where x is measured in meters and t is in seconds, findThe average velocity between t=1 s and 2 s.The average acceleration between t=1 s and t=2 s.Know the difference between average and instantaneous quantities. Be able to calculate average velocity/acceleration, or instantaneous velocity/ acceleration given equations as a function of time.The instantaneous velocity at 1 second.The instantaneous acceleration at 1 second. If the position of a particle is given by x=20t-5t3, where x is in meters and t is in seconds. a. when, if ever, is the particle’s velocity zero?b. When is its acceleration a zero?Given an expression for one of the kinematic quantities, position, velocity, or acceleration, as a function of time, determine the other two as a function of time (derivatives/antiderivatives) The position function x(t) of a particle moving along an x axis is x=4.0-6.0t2, with x in meters and t in seconds.At what time does the particle momentarily stop?Where does the particle momentarily stop?Graph x versus t What is the derivative of a sin(x) graph? Now let’s practice an antiderivative (integral)What is the power rule to find area instead of slope? How do we symbolize that we are finding area? Remember area is the CHANGE of a quanitity.Given an expression for one of the kinematic quantities, position, velocity, or acceleration, as a function of time, determine the other two as a function of time (derivatives/antiderivatives)Now find the area of the velocity vs time graph if acceleration is constant.Given a=5t4-3t2+2t use your graphing calculate to find the change of velocity from 2 to 6 seconds. An object moving in a straight line has a velocity v in meters per second that varies with time t in seconds according to the following function.v=4+0.5t2 The instantaneous acceleration of the object at t=2 seconds is2m/s24m/s25m/s26m/s28m/s2 The displacement of the object between t=0 and t=6 seconds is22m28m40m42m60mA particle of mass m moves along a straight path with a speed v defined by the function v=bt2+c, where b and c are constants and t is time. What is the magnitude F of the net force on the particle at time t=t1? ( F=ma)bt12+c3mbt1+2cmbt1mbt1+c2mbt1A particle moves along the x axis with a nonconstant acceleration described by a=12t, where a is in meters per second squared and t is in seconds. If the particle starts from rest so that its speed v and position x are zero when t=0, where is it located when t=2 seconds?x=12mx=16mx=24mx=32mx=48mThe velocity v in meters per second of an object moving in a straight line is given as a function of time t in seconds by v=4t3 +2t. The total distance the object travels between t=1s and t=2 s is12 m18 m24 m30 m36 mGiven an expression for one of the kinematic quantities, position, velocity, or acceleration, as a function of time, determine the other two as a function of time (derivatives/antiderivatives)As a drag car moves from rest at time t=0, its velocity varies as the square of the elapsed time according to the equation v=bt2, where b is a constant. The expression for the distance traveled by the car from its position at t=0 is3838575112395bt3bt3/34bt23bt2bt2/3 A particle moves along a straight line. Its speed in m/y is given by v=a+bt2, where a and b are constants and t is time in seconds. How far does the particle move between t=1 s and t=2s?2ba+ba+2ba+(7/3) ba+ (7/2) b ................
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