3.2 Motion with Constant Acceleration Essential Questions 1. What do a position-time graph and a velocity-time graph look like for motion with constant acceleration? 2. How can you determine the displacement of a moving object from its velocity-time graph? 3. What are the relationships among position, velocity, acceleration, and time? Vocabulary Review displacement Copyright © McGraw-Hill Education Motion with Constant Acceleration

A. Position with Constant Acceleration 1. The table and graph shows position data for a car with constant acceleration. 2. The graph shows that the car’s motion is not uniform: The displacements for equal time intervals on the graph get larger and larger. 3. The slope of a position-time graph of a car moving with a constant acceleration gets steeper as time goes on. Copyright © McGraw-Hill Education Motion with Constant Acceleration

A. Position with Constant Acceleration 4. For an object with constant acceleration, the position-time graph is a parabola. 5. The slopes from the position time graph can be used to create a velocity-time graph. 6. A unique position-time graph cannot be created using a velocity-time graph because it does not contain any information about the object’s position. However, the area under the v-t graph is equal to the object’s displacement. Motion with Constant Acceleration Copyright © McGraw-Hill Education

Practice Problem 20. The graph in Figure 13 describes the motion of two bicyclists, Akiko and Brian, who start from rest and travel north, increasing their speed with a constant acceleration. What was the total displacement of each bicyclist during the time shown for each? Hint: Use the area of a triangle: area = 1/2(base)(height).

Practice Problem 21. The motion of two people, Carlos and Diana, moving south along a straight path is described by the graph in Figure 14. What is the total displacement of each person during the 4.0-s interval shown on the graph?

B. Velocity with Average Acceleration 1. If an object’s average acceleration during a time interval is known, then it can be used to determine how much the velocity changed during that time. 2. In cases in which the acceleration is constant, the average acceleration (ā) is the same as the instantaneous acceleration (a). 3. Rearrange the formula to find the time when the object has constant acceleration: 4. Rearrange the formula to find the initial velocity: Final Velocity with Average Acceleration Copyright © McGraw-Hill Education Motion with Constant Acceleration

Practice Problem 17. A bus traveling 30.0 km/h east has a constant increase in speed of 1.5 m/s2. What is its velocity 6.8 s later? Given: Formula: Work: Answer:

Practice Problem 18. If a car accelerates from rest at a constant rate of 5.5 m/s2 north, how long will it take for the car to reach a velocity of 28 m/s north? Given: Formula: Work: Answer:

Position with Average Acceleration B. Velocity with Average Acceleration 5. The graph describes constant acceleration that started with an initial nonzero velocity. 6. To determine the displacement, you can divide the area under the graph into a rectangle and a triangle. The total area is then: 7. If ti = 0, the equation then becomes the following: Position with Average Acceleration Copyright © McGraw-Hill Education Motion with Constant Acceleration

Practice Problem 24. A race car travels on a straight racetrack with a forward velocity of 44 m/s and slows at a constant rate to a velocity of 22 m/s over 11 s. How far does it move during this time? Given: Formula: Work: Answer:

C. An Alternate Equation 1. Often, it is useful to relate position, velocity, and constant acceleration without including time. 2. Rearrange the velocity with average equation to solve for tf = (vf − vi)/a. 3. Substitute into the equation for position with average acceleration equation and rearrange. Velocity with Constant Acceleration Copyright © McGraw-Hill Education Motion with Constant Acceleration

Practice Problem vi D. Joni jogs at a velocity of 2.50 m/s. If she then accelerates at a constant −0.10 m/s2, how fast will she be jogging when she has moved 10.0 m? Given: Formula: Work: Answer: a +x Copyright © McGraw-Hill Education Motion with Constant Acceleration

Practice Problem Part 1 Part 2 a E. A cat runs at 2.0 m/s for 3.0 s, then slows to a stop with an acceleration of −0.80 m/s2. What is the cat’s displacement during this movement? Given: Formula: Work: Answer: vi vi +x +x Motion with Constant Acceleration Copyright © McGraw-Hill Education

Practice Problem 25. A car accelerates at a constant rate from 15 m/s to 25 m/s while it travels a distance of 125 m. How long does it take to achieve the final speed? Given: Formula: Work: Answer:

3.2 Assignment: