Average and Instantaneous Acceleration

Average and Instantaneous Acceleration by SHS Encoder 3 on November 24, 2017 lesson duration of 20 minutes under General Physics 1 generated on November 24, 2017 at 01:23 pm Tags: Instantaneous Acceleration, Average

Generated: Nov 24,2017 09:23 PM Average and Instantaneous Acceleration ( 1 hour and 20 mins ) Written By: SHS Encoder 3 on June 29, 2016 Subjects: General Physics 1 Tags: Instantaneous Acceleration, Average Resources University Physics with modern Physics (12th ed.) Young, H. D., & Freedman, R. A. (2007). University Physics with modern Physics (12th ed.). Boston, MA: Addison- Wesley. Physics (4th ed.) Resnick, D., Halliday, R., & Krane, K. S. (1991). Physics (4th ed.). Hoboken, NJ: John Wiley & Sons. Content Standard The learners demonstrate an understanding of... 1. Position, time, distance, displacement, speed, average velocity, instantaneous velocity 2. Average acceleration, and instantaneous acceleration 3. Uniformly accelerated linear motion 4. Free-fall motion 5. 1D Uniform Acceleration Problems Performance Standard The learners are able to solve, using experimental and theoretical approaches, multiconcept, rich-context problems involving measurement, vectors, motions in 1D, 2D, and 3D, Newton s Laws, work, energy, center of mass, momentum, impulse, and collisions Learning Competencies The learners recognize whether or not a physical situation involves constant velocity or constant acceleration The learners interpret velocity and acceleration, respectively, as slopes of position vs. time and velocity vs. time curves 1 / 7

The learners construct velocity vs. time and acceleration vs. time graphs, respectively, corresponding to a given position vs. time-graph and velocity vs. time graph and vice versa INTRODUCTION/MOTIVATION 5 mins 1. Do a quick review of the previous lesson on displacement, average velocity and instantaneous velocity. INSTRUCTION 20 mins 1. The acceleration of a moving object is a measure of its change in velocity. Discuss how to calculate the average acceleration from the ratio of the change in velocity to the time duration of this change. 2. Recall that the first derivative of the displacement with respect to time is the instantaneous velocity. Discuss that the instantaneous acceleration is the first derivative of the velocity with respect to time: Figure 1: Average acceleration 3. Thus, given the displacement as a function of time, the acceleration can be calculated as a function of time by successive derivations: 2 / 7

4. Given a constant acceleration, the change in velocity (from an initial velocity) can be calculated from the constant average velocity multiplied by the time interval. Figure 3: Velocity as area under the acceleration versus time curve Special case: motion with constant acceleration Derive the following relations (for constant acceleration): From Eqn 2 and Eqn 3, the total displacement (from an initial position to a final position) can be derived as a function 3 / 7

of the total time duration (from an initial time to a final time) and the constant acceleration: 5. Discuss that with a time-varying acceleration, the total change in velocity (from an initial velocity) can be calculated as the area under the acceleration versus time curve (at a given time duration). Given a constant acceleration (Figure 3), the velocity change is defined by the rectangular area under the acceleration versus time curve subtended by the initial and final time. Thus, with a continuously time-varying acceleration, the area under the curve is approximated by the sum of the small rectangular areas defined by the product of small time intervals and the local average acceleration. This summation becomes an integral when the time duration increments become infinitesimally small. DISCUSSION 20 mins 1. Review the relations between displacement and velocity, velocity and acceleration in terms of first derivative in terms of time and area under the curve within a time interval. 2. Discuss how one can identify whether a velocity is constant (zero, positive or negative), time varying (slowing down or increasing) using Figure 4. 3. Replace the displacement variable with velocity in Figure 4 (Figure 5) and discuss what the related acceleration becomes (constant or time varying). 4. Discuss the inverse: deriving the shape of the displacement curve based on the velocity versus time graph; deriving the shape of the velocity curve based on the acceleration versus time graph. Summary: Displacement versus time: Graph of a line with positive/negative slope à positive/negative constant velocity Graph with monotonically increasing slope à increasing velocity Graph with monotonically decreasing slope à decreasing velocity 4 / 7

Velocity versus time: Graph of a line with positive/negative slope à positive/negative constant acceleration Graph with monotonically increasing slope à increasing acceleration Graph with monotonically decreasing slope à decreasing acceleration Warning: the non-linear parts of the graph were strategically chosen as sections of a parabola hence, the corresponding first derivate of these sections is either a negatively sloping line (for a downward opening parabola) or a positively sloping line (for an upward opening parabola) Figure 4: Displacement versus time and the corresponding velocity graphs Figure 5: Velocity versus time and the corresponding acceleration graphs DISCUSSION 20 mins 1. Review the relations between displacement and velocity, velocity and acceleration in terms of first derivative in terms of time and area under the curve within a time interval. 2. Discuss how one can identify whether a velocity is constant (zero, positive or negative), time varying (slowing down or increasing) using Figure 4. 3. Replace the displacement variable with velocity in Figure 4 (Figure 5) and discuss what the related acceleration becomes (constant or time varying). 4. Discuss the inverse: deriving the shape of the displacement curve based on the velocity versus time graph; deriving the shape of the velocity curve based on the acceleration versus time graph. Summary: Displacement versus time: 5 / 7

Graph of a line with positive/negative slope à positive/negative constant velocity Graph with monotonically increasing slope à increasing velocity Graph with monotonically decreasing slope à decreasing velocity Velocity versus time: Graph of a line with positive/negative slope à positive/negative constant acceleration Graph with monotonically increasing slope à increasing acceleration Graph with monotonically decreasing slope à decreasing acceleration Warning: the non-linear parts of the graph were strategically chosen as sections of a parabola hence, the corresponding first derivate of these sections is either a negatively sloping line (for a downward opening parabola) or a positively sloping line (for an upward opening parabola) Figure 4: Displacement versus time and the corresponding velocity graphs Figure 5: Velocity versus time and the corresponding acceleration graphs Enrichment/Evaluation 15 mins Option for group discussions 1. Given a sinusoidal displacement versus time graph (displacement = A sin(bt bt); b = 4?/s, A = 2 cm), ask the class to graph the corresponding velocity versus time and acceleration versus time graphs. Recall that the velocity is the first derivative of the displacement with respect to time and that the acceleration is the first derivative with respect to time. At which parts of the graph would the velocity or acceleration become zero or at maximum value (positive or negative)? Discuss where the equilibrium position would be based on the motion (as illustrated by the displacement versus curve graph). What happens to the velocity and acceleration at the equilibrium position? Download Teaching Guide Book 0 mins 6 / 7