Chapter 2 – Motion in One Dimension Term Frame of Reference Position Distance Displacement Definition Example A point of origin for describing motion; serves as a designated starting point for defining motion. The railroad tracks The separation between an object and the frame of reference; precisely describes the location of an object. The car is currently located 200 m east of the railroad tracks. A measure of how much ground an object has covered; tells us how far we are actually moving. The truck is 125 m from the tracks. The change in position of an object; tells us where we are located relative to the starting point. Since crossing the tracks, the car has moved 200 m to the east. Symbol Unit Mathematical Relationship 2 Term Average Speed Average Velocity Instantaneous Velocity Definition Example The average rate at which an object covers ground; a measure of how fast an object is moving along its actual path; defined as distance per unit time. The average rate at which an object changes its position; a measure of how fast and in what direction an object is moving away from its starting point; defined as displacement per unit time. The velocity of an object at some particular instant. The average speed of the car during its trip was 50 mph (22.2 m/s). The average velocity of the truck during its trip was 35 mph (15.6 m/s) to the east. At one particular instant, the car was moving at 40 mph (17.8 m/s) to the east. Symbol Unit Mathematical Relationship 3 Term Average Acceleration Instantaneous Acceleration Definition Example The average rate at which an object changes its velocity; a measure of how fast and in what direction the velocity is changing. The average acceleration of the car is 4.5 mph/s (2 m/s2) to the east. The acceleration of an object at some particular instant. At one particular instant, the acceleration of the car was 2.25 mph/s (1 m/s2) to the east. Symbol Unit Mathematical Relationship 4 One-Dimensional Motion Example Problems 1. A car drives 40 miles east, turns around, and then drives 30 miles west. (a) Find the distance traveled by the car. (b) Find the car’s displacement. 2. Assuming that the trip described in Problem 1 takes 2 hours, determine the following: (a) The average speed of the car. (b) The average velocity of the car. 3. A bus travels 280 km south along a straight path with an average velocity of 80 km/h to the south. The bus stops for a 30 minute break, then travels 140 km south with an average velocity of 70 km/h to the south. (a) How long does the total trip last? (b) What is the average velocity for the total trip? 4. A car traveling at 7 m/s accelerates uniformly at 2.5 m/s2 to reach a speed of 12 m/s. How long does it take for the acceleration to occur? 5 Acceleration, Velocity, Displacement, & Time Relationships Chapter 2 Worksheet 1. A battery-operated vehicle travels with a constant velocity of 30 m/s over a 10 s time interval. Complete the table below, and then use this data to answer the questions that follow: Time Elapsed (s) Total Displacement (m) Velocity (m/s) 0 0 30 1 30 2 30 3 30 4 30 5 30 6 30 7 30 8 30 9 30 10 30 Acceleration (m/s2) (a) Plot a graph of velocity vs. time. What is the shape of the graph? By examining your acceleration data, how could you have predicted the shape of this graph? What is the slope of the graph? How does the slope of this graph relate to the acceleration of the vehicle? What is the area under the graph? How does the area under the graph relate to the displacement of the vehicle over this time interval? Velocity (m/s) Velocity vs. Time 40 30 20 10 0 0 1 2 3 4 5 Time (s) 6 7 8 9 10 6 (b) Plot a graph of total displacement vs. time. What is the shape of the graph? By examining your graph of velocity vs. time, how could you have predicted the shape of this graph? What is the slope of the graph? How does the slope of the graph compare with the velocity of the vehicle over the time interval? Displacement (m) Total Displacement vs. Time 300 240 180 120 60 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) (c) Using only the velocity and time values, how could one determine the total displacement of the cart after any length of time? (d) Assuming this motion was to continue indefinitely, how far would the cart have traveled after 100 s? 7 2. A motorized car accelerates at a constant rate of 10 m/s2 over a time interval of 10 s. Complete the table below, and then use this data to answer the questions that follow: Time Elapsed (s) Total Displacement (m) Velocity (m/s) Acceleration (m/s2) 0 0 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 (a) Plot a graph of acceleration vs. time. What is the shape of the graph? What is the slope of the graph? What does the slope of the graph tell you about the motion of the car? What is the area under the graph? What is the significance of the area under the graph? Acceleration (m/s2) Acceleration vs. Time 15 10 5 0 0 1 2 3 4 5 Time (s) 6 7 8 9 10 8 (b) Plot a graph of velocity vs. time. What is the shape of the graph? By examining your graph of acceleration vs. time, how could you have predicted the shape of this graph? What is the slope of the graph? What is the significance of the slope of the graph? What is the area under the graph? What is the significance of the area under the graph? Velocity (m/s) Velocity vs. Time 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) (c) Plot a graph of total displacement vs. time. What is the shape of the graph? By examining your graph of velocity vs. time, how could you have predicted the shape of this graph? Find the slope of the graph over the time interval from 0 to 10 s. What is the significance of the slope of the graph in terms of the car’s velocity over that interval? Displacement (m) Total Displacement vs. Time 500 400 300 200 100 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) (d) Using only the given acceleration, initial and final velocity values, and amount of elapsed time, how could one determine the total displacement of the cart after any length of time? (e) Assuming this motion was to continue indefinitely, how far would the cart have traveled after 100 s? 9 Displacement, Velocity, & Acceleration from Graphs Chapter 2 Worksheet 1. The slope of a position-time or total displacement-time graph over any interval of time gives the _______________ _______________ during that interval. 2. The tangent to a position-time or total displacement-time graph at any point gives the _______________ _______________ at that point. 3. The graph below shows the total displacement of an object from a given starting point over a time interval of 20 seconds. Use the graph to answer the questions that follow: Position (m) Position vs. Time 20 15 10 5 0 0 5 10 15 20 Time (s) (a) What is the object’s position at t = 10 s? (b) How far did the object travel between t = 5 s and t = 15 s. (c) What is the object’s displacement between t = 15 s and t = 20 s? (d) What is the total displacement of the object over the entire 20 s time interval? (e) Find the average velocity of the object between t = 5 s and t = 10 s. (f) What is the instantaneous velocity of the object at t = 7 s. (g) Describe the motion of the object between t = 10 and t = 15 s. 10 4. The slope of a velocity-time graph over any interval of time gives the _______________ _______________ during that interval. 5. The tangent to a velocity-time graph at any point gives the _______________ _______________ at that point. 6. The area under a velocity-time graph between any two times gives the _______________ during that interval. 7. The graph below shows the velocity of an object over a time interval of 40 seconds. Use the graph to answer the questions that follow: Velocity (m/s) Velocity vs. Time 12 8 4 0 0 5 10 15 20 25 30 35 40 Time (s) (a) What is the velocity of the object at t = 15 s? (b) Describe what is happening to the velocity of the object at t = 25 s? (c) Find the average acceleration of the object between t = 0 and t = 5 s. (d) Find the average acceleration of the object between t = 5 s and t = 30 s. (e) Find the average acceleration of the object over the entire 40 s time interval. (f) Find the instantaneous acceleration of the object at t = 35 s. (g) How far did the object travel between t = 5 s and t = 15 s? (h) What is the object’s displacement between t = 15 s and t = 20 s? (i) Find the displacement of the object over the entire 40 s time interval. 11 8. The area under an acceleration-time graph between any two times gives the ______________________________ during that interval. 9. The graph below shows the acceleration of an object over a time interval of 8 seconds. Use the graph to answer the questions that follow: Acceleration 2 (m/s ) Acceleration vs. Time 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 Time (s) (a) What is the object’s instantaneous acceleration at t = 5 s? (b) What is the change in acceleration of the object between t = 3 s and t = 7 s? (c) Describe what is happening to the velocity of the object between t = 4 s and t = 6 s. (d) What is the object’s change in velocity between t = 0 s and t = 4 s? 12 Equations of Motion for Uniform Acceleration Uniform or Constant Acceleration is acceleration that does not change with time. The velocity-time graph of a body experiencing such motion is a straight line. The following equations are valid only for such cases: (1) (2) (3) (4) Equation: vf = vi + at d = ½ (vf + vi)t d = vit + ½ at2 vf2 = vi2 + 2ad Relates: (must know 3 of the 4) Initial Velocity Final Velocity Acceleration Time Displacement Initial Velocity Final Velocity Time Displacement Initial Velocity Acceleration Time Initial Velocity Final Velocity Acceleration Displacement Not Needed: Displacement Acceleration Final Velocity Time Since aavg = v and t v = vf – vi, v vi a= f . t Derivation: Solving for vf, we obtain the result vf = vi + at. Previously, we found that d = vavgt. Earlier we found that vf = vi + at. And, for an object experiencing constant acceleration, vavg = ½ (vf + vi). Substituting this for vf in eqn (2), d = ½((vi+at)+vi)t = ½ (2vi+at)t. Combining these, d = ½ (vf + vi)t. Simplifying, d = vit + ½ at2. Solving eqn (1) for t, we obtain v vi t = f . a Now substituting this into eqn (2), v vi v f v i d = f 2 a = v f 2 vi 2 . 2a And rearranging, vf2 = vi2 + 2ad. 13 Acceleration Due to Gravity Chapter 2 Worksheet A ball is fired up it the air with an initial velocity of 49 m/s, and its velocity and displacement is monitored over a 10 s time interval. Complete the table below, and then use this data to answer the questions that follow: Time Elapsed (s) Total Displacement (m) Velocity (m/s) Acceleration (m/s2) 0 0 49 –9.8 1 –9.8 2 –9.8 3 –9.8 4 –9.8 5 –9.8 6 –9.8 7 –9.8 8 –9.8 9 –9.8 10 –9.8 (a) Plot a graph of acceleration vs. time. What is the shape of the graph? What is the slope of the graph? What is the significance of the slope of the graph? What is the area under the graph? What is the significance of the area under the graph? Acceleration (m/s2) Acceleration vs. Time 20 10 0 -10 -20 0 1 2 3 4 5 Time (s) 6 7 8 9 10 14 (b) Plot a graph of velocity vs. time. What is the shape of the graph? By examining your graph of acceleration vs. time, how could you have predicted the shape of this graph? What is the slope of the graph? What is the significance of the slope of the graph? What is the area under the graph? What is the significance of the area under the graph? Velocity (m/s) Velocity vs. Time 50 25 0 -25 -50 0 1 2 3 4 5 6 7 8 9 10 Time (s) (c) Plot a graph of total displacement vs. time. What is the shape of the graph? By examining your graph of velocity vs. time, how could you have predicted the shape of this graph? Find the slope of the graph over the time interval from 0 to 5 s. What is the significance of the slope of the graph in terms of the ball’s velocity over that interval? Displacement (m) Total Displacement vs. Time 125 100 75 50 25 0 0 1 2 3 4 5 Time (s) 6 7 8 9 10 15 Acceleration Due to Gravity Problem: A ball is thrown up into the air with an initial speed of 20 m/s from the top of a bridge located 30 m above the surface of the river below. (a) How long will it take the ball to reach the top of its path? (b) Relative to the bridge, how high will the ball rise? (c) From the moment of its release, how long will it take the ball to hit the water? (d) How fast will the ball be moving when it strikes the water? 16 Chapter 2 Worksheet – Review of Motion in One Dimension 1. The following graph shows the velocity of a moving object as monitored over a time period of 8 s. Use the graph to answer the questions that follow: Velocity (m/s) Velocity vs. Time 4 2 0 -2 0 1 2 3 4 5 6 7 8 Time (s) (a) Determine the distance traveled by the object between t = 0 s and t = 8 s. (b) Determine the displacement of the object between t = 0 s and t = 8 s. (c) What was change in the object’s velocity between t = 2 s and t = 6 s? (d) Find the average acceleration of the object between t = 3 s and t = 8 s. (e) Calculate the average acceleration of the object over the entire 8 s interval. 2. The VW Beetle goes from 0 to 60 mph with an acceleration of 2.35 m/s2. (a) Using the fact that there are 1.6 km/mile, convert the final velocity to m/s. (b) Starting from rest, how many seconds should it take the VW Beetle to reach this final velocity? (c) A dragster can go from 0 to 60 mph in a mere 0.600 s. What is the acceleration rate (in m/s2) of the dragster? 3. A golf ball is dropped from rest into a river from a bridge 55 m above the water. A short time later, a second ball is thrown downward with a speed of 11.9 m/s, and happens to strike the water at the same moment as the first ball. (a) How long did it take the first ball to reach the water? (b) How long was the first ball falling before the second was thrown? (c) With what velocity did the first ball strike the water? (d) With what velocity did the second ball strike the water? 4. A woman on a bridge 100 m high sees a raft floating at a constant speed on the river below. She drops a stone from rest and is successful in hitting the raft. The stone is released when the raft has 6 m more to travel before passing under the bridge. (a) How long does it take the stone to reach the water? (b) At what constant speed is the raft traveling? Answers: (1) 14 m, 6 m, –6 m/s, –0.8 m/s2, 0, (2) 26.7 m/s, 11.4 s, 44.5 m/s2, (3) 3.35 s, 1.00 s, –32.8 m/s, –34.9 m/s, (4) 4.52 s, 1.33 17

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# Chapter 2 – Motion in One Dimension