1. How could you determine the average speed of an object whose motion is represented in the graphs above?

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1 AP Physics Lesson 1 b Kinematics Graphical Analysis and Kinematic Equation Use Outcomes Interpret graphical evidence of motion (uniform speed & uniform acceleration). Apply an understanding of position time graphs to novel examples. Use graphical patterns of uniform velocity and uniform acceleration to derive kinematic equations. Solve graphical analysis problems and problems involving kinematic equations. Name Date Period Engage A. B. C. 1. How could you determine the average speed of an object whose motion is represented in the graphs above? 2. From the previous lesson we know that not all three objects are moving at a uniform velocity, how could you determine the velocity of each object at the end of the time interval shown? Share your answers with your lab group members. Discuss your answers with your lab group. Notes Vocabulary List: average speed, average velocity, acceleration, energy, dynamics, distance, displacement, four, force, instantaneous speed, instantaneous velocity, kinematics, linear, mass, momentum, scalar, slope, temperature, vector, speed, velocity, v=d/t, A. : The study and description of motion of objects. B. : The study of the causes of motion (we ll investigate this later) C. Quantities: Measurements that have a magnitude (size) but no direction. a. ex. c. ex. b. ex. d. ex. D. : Total path length covered in moving from location to another. E. : The total path length covered in a given amount of time. a. equation: V= d = distance traveled total time to travel that distance = d t 2 t 1 b. SI units (international system of units) are m/s. F. : The speed of an object at a particular moment in time. G. Quantities: Measurements that have a magnitude and direction. a. ex. b. ex. 1

2 H. : Distance in a particular direction. a. ex. b. Practice: imagine that this person walked from the door to the girl and back to the water fountain. What would the displacement be? I. : The displacement divided by the total travel time. a. = x 2 x 1 = x x o t 2 t 1 t t o ( ) ( ( ) = x x o ) ( t) total displacement = total time b. The average slope of the displacement time graph. 2

3 J. : The velocity of an object at a particular instant in time. a. The of the displacement vs. time line. b. Occurs when the time interval shrinks to zero. c. Represented by a tangent line drawn to the displacement vs. time line. II. Graphical Analysis Practice A. Calculate the average velocities of the objects represented by these graphs B. Calculate the average velocities for these objects for t= 0-2 seconds and for t= 0-4 seconds C. Determine the instantaneous velocities for graphs 2 and 3 at 2 seconds and at 4 seconds. v 2= v 4= 3

4 A. Examine the graph below. 1. What kind of motion is this object moving with? 2. What is the slope of this line at 2 seconds? 3. What is the slope of this line at 4 seconds? 4. What is the slope of this line at 6 seconds? 5. In the space below, sketch the appearance of a velocity vs. time graph for this data. III. Using AREAS to predict displacement values. A. Examine the graph below. 1. What kind of motion is this object moving with? 2. What is the area under this line at 2 seconds? (Remember that the area under a triangle is a= 1 2 b h ) 3. What is the area under this line at 4 seconds? 4. What is the area under this line at 6 seconds? 5. In the space above, sketch the appearance of a displacement vs. time graph for this data. 4

5 IV. Kinematic Equations 1. The area of a rectangle a=l w (area = length x width) has the same format as a equation y=kx. 2. The kinematic equations listed below are all derived from the slopes and areas of graphs!. For uniform velocity the acceleration is zero ( a=0 ) and the equations above will simplify accordingly. v and x are the final values whereas v 0 and x 0 are the initial values. III. : Rate of change of velocity. a. a = Δv = v 2 v 1 t 2 t 1 = v v o t t o b. Important note! For uniformly accelerated objects only v final =2 v (Final instantaneous velocity equals 2 times the average velocity minus the initial velocity, but when the initial velocity is zero, this becomes v final =2 v ) IV. Graphical Analysis of Motion Practice A. Plot the values and use slopes to generate the average velocity and acceleration graphs. 1. Uniform Velocity Velocity (m/s)

6 2. Uniform Acceleration Average Velocity from start (m/s) Final or Instantaneous Velocity v final =2 v Acceleration a = Δv B. Plot the values and use areas to generate the displacement vs. time graphs. 1. Uniform Velocity Velocity from start (m/s)

7 x= 1 2 v t or x= 1 2 a t 2 2. Uniform Acceleration Final or Instantaneous Velocity v=a t Acceleration V. Yeah!!!! More Practice C. Determine the values for the velocity vs. time table and graph the values. Velocity from start (m/s) D. Average Velocity from start (m/s) Final or Instantaneous Velocity v final =2 v Acceleration a = Δv

8 E. Velocity from start (m/s) F. x= 1 2 v t or x= 1 2 a t 2 Final or Instantaneous Velocity v=a t Acceleration

9 G. Mixed Motion use slopes and areas to complete the graphs below. You can t use the kinematic equations since the motion is mixed. You have to treat each segment separately. This is an example where using the graphs and slopes and areas is real helpful. Final or Instantaneous Acceleration Velocity VI. Conceptual Sketches. 1. Prepare sketches that match with the provided graphs. Assume all initial displacements=0. a. 9

10 Explore II In this section you will use the kinematic equations derived through graphical analysis to solve problems. The teacher will model how to solve an example problem on the left side, you will solve a similar problem on the right side. Uniform Velocity Equations Uniform Acceleration Equations x= x Δv o +vt a= v 2 =v 2 o +2ax v f = v o + at x= x o +v o t at 2 Problem Solving Strategy (modified 7-step method we used in 9 th grade) Example: A toy car accelerates over a distance of 3m at a rate of 3m/s 2 to a final velocity of 12m/s. What was the initial velocity of the toy car? 1. State the major concept used to solve the problem e.g. Uniform Acceleration 2. Record ALL relevant information and equations e.g. x=3m, a=3m/s 2, v f=12m/s x= x o +v o t at 2 3. Perform all algebraic manipulations BEFORE substituting in numerical values! e.g. v f 2 =v o 2 +2ax v f 2 2ax=v o 2 v f 2 2ax =v o 4. Substitute in numerical values and solve. v 2 f 2ax =v o e.g. (12m / s) 2 2(3m / s 2 )(3m)=v o =11.2m / s SHOW CORRECT UNITS THROUGHOUT! 5. your answer! Practice Problems The toy car on the left travels at a uniform velocity of 15 cm/s over a displacement of 90cm. Pay attention to the units. Standard units are [m] for displacement and [m/s] for velocity. 1. How much time does it take the car to cover the total distance? Step 1. Step 2. Step 3. The kick dis hoverpuck travels at a uniform velocity of 20 cm/s over a displacement of 500cm. 2. How much time does it take the hoverpuck to cover the total distance? Step 1. Step 2. Step 3. Step 4. Step 4. Step 5. Step 5. 10

11 An object moves at a uniform velocity for 2 seconds, then instantaneously increases to new uniform velocity. An object moves at a uniform velocity for 3 seconds, then instantaneously decreases to new uniform velocity. 3. What is the average velocity for the entire 7 second interval? 6. What is the average velocity for the entire 7 second interval? 4. What is the velocity of the object for the 0-2 and 2-7 second intervals? 7. What is the velocity of the object for the 0-3 and 3-7 second intervals? 5. Use the information in the graph above to complete the velocity time graph below. 8. Use the information in the graph above to complete the velocity time graph below. An object accelerates uniformly from rest to a final velocity of 15 m/s in 3 seconds. 9. What is the acceleration rate of the object? An object accelerates uniformly from rest to a final velocity of 120 m/s in one minute. 13. What is the acceleration rate of the object? 10. What is the total displacement of the object at the end of 3 seconds? 14. What is the total displacement of the object at the end of one minute? 11. What is the average velocity of the object over the entire 3 seconds? 15. What is the average velocity of the object over the entire 60 seconds? 12. Sketch the following graphs for this event. 16. Sketch the following graphs for this event. 11

12 17. What is the acceleration rate of the dragster? A dragster traveling at 70 m/s deploys a parachute and experiences a uniform deceleration over a distance of 200 m and achieves a final velocity of 5m/s. 19. What is the acceleration rate of the dragster? 18. How much time does it take the dragster to cover the 400m distance? 20. How much time does it take the dragster to cover the 200m distance? A dragster accelerates uniformly from rest over a distance of 400 meters. The dragster achieves a final velocity of 73 m/s. 21. Sketch the following graphs for this event. A ball rolling at 2m/s encounters a ramp and experiences a uniform negative 2 acceleration of -0.9 m/s for 3.0 seconds while rolling on the ramp. 25. How far up (not how high we can t do that yet) the ramp does the ball travel in this time? A boat initially traveling at 10. m/s accelerates uniformly at the rate of 5 2 m/s for 10. seconds. 22. How far does the boat travel during that time? 23. What is the final velocity of the boat after 10. seconds? 26. What is the final velocity of the ball after 3.0 seconds? 27. At what time will the ball have a final velocity of 0m/s? 24. Sketch the following graphs for this event. 28. Sketch the following graphs for this event. 12