Chapter 2: Motion Along A Straight Line 963612

Chapter 2 Motion Along A Straight Linedue 06092017 Noonplease Show

Analyze and solve problems related to motion along a straight line, including concepts of position, displacement, speed, velocity, acceleration, free fall, and uniform acceleration. Show all calculations clearly, indicating units and significant figures. Draw relevant graphs where appropriate, and interpret given data to find answers to various kinematic questions.

Paper For Above instruction

Understanding the fundamental principles of motion along a straight line is crucial in physics as it forms the basis for analyzing real-world movements. This paper addresses various kinematic concepts by solving different problems involving displacement, speed, velocity, acceleration, and free fall, emphasizing clarity and detailed reasoning in each step to facilitate comprehensive understanding.

Position and Displacement

The first problem involves a car's motion observed at various times, providing position data relative to its starting point. At 3:00 p.m., the car is 20 km south of its initial point; at 4:00 p.m., it is 96 km farther south; at 6:00 p.m., it is 12 km south of the starting point. The task involves calculating displacements between specific times and from the starting point.

Displacement between 3:00 p.m. and 6:00 p.m.: The difference in position is from 20 km south to 12 km south of the starting point. Moving from 20 km south to 12 km south involves a net northward displacement of 8 km. Since displacement is a vector quantity, its sign indicates direction; here, the net displacement is 8 km north.

Displacement from start to 4:00 p.m.: The position at 4:00 p.m. is 96 km south, hence the displacement from the initial point (at 3:00 p.m.) to 4:00 p.m. is 96 km south, or -96 km if considering south as negative direction.

Displacement between 4:00 p.m. and 6:00 p.m.: The position shifts from 96 km south to 12 km south of the starting point. The net change is 84 km north, indicating a displacement of 84 km north.

Speed and Velocity

Calculations involve average speed and velocity for different trip segments. For Joe’s trip: Traveling 1.5 km in 20 minutes (1200 seconds) and returning in 30 minutes (1800 seconds) involves converting times to seconds and distances to meters to find average speed in m/s.

Average speed for entire trip: Total distance traveled divided by total time, considering both outbound and return trips, yields an overall average speed. The total distance is (1.5 km + 1.5 km) = 3 km or 3000 meters, and total time is 1200 s + 1800 s = 3000 s. The average speed is 3000 m / 3000 s = 1.0 m/s.

For the race car traveling north: The distance is 750 m in 20 s and 750 m in 25 s upon return, totaling 1500 m over (20 s + 25 s) = 45 s. Average speed is total distance divided by total time, i.e., 1500 m / 45 s ≈ 33.33 m/s. The average velocity vector considers initial and final positions, but since the car ends where it started, the net displacement is zero, and thus average velocity is zero.

Jogging around a circular track of 300 m diameter: The total distance jogged is multiple laps, but if only one lap in 10 minutes (600 seconds), then average speed = total distance / time. The magnitude of average velocity considers the net displacement after completing laps; since the jogger ends where they started, the net displacement is zero, making the average velocity zero.

Jason’s trip: Total distance covered on the way west (with different speeds) and initial and final velocities are used to find average velocity vector for the entire trip, accounting for the combined displacement over total time.

Further problems involve computing displacement over time, interpreting velocity graphs, and analyzing motion parameters from graph data, emphasizing the importance of understanding position, velocity, and acceleration relationships in kinematics.

Acceleration

Problems include calculating average acceleration during braking and acceleration phases, involving initial and final velocities and elapsed time. For example, a car decelerating from 28 m/s to stop in 4 seconds has an average acceleration: a = (final velocity - initial velocity) / time = (0 - 28 m/s) / 4 s = -7 m/s², indicating deceleration.

Similarly, an airplane accelerating from rest to 35 m/s in 8 s has an average acceleration of (35 m/s - 0) / 8 s ≈ 4.375 m/s².

Graphical representation of acceleration over time is essential in understanding how acceleration relates to motion; constant acceleration graphs are linear, whereas variable acceleration graphs are non-linear.

Free Fall

Equations of motion under gravity are used to determine fall times, velocities, and displacements of objects in free fall. For a golf ball falling from rest over 12.0 m: Time can be found using s = ½ g t², which yields t = √(2s / g). Substituting s = 12 m and g ≈ 9.8 m/s², t ≈ 1.56 s. For twice the distance, the fall time increases by √2, approximately 2.21 s.

Grant’s jump: Using initial velocity v₀ and displacement to find initial speed, applying s = v₀ t + ½ g t², allows calculating the initial speed as v₀ = s / t + ½ g t, which gives approximately 5.44 m/s.

The camera dropped from a cliff: The free fall equations, considering upward or downward motion and initial speeds, help in analyzing different scenarios, including horizontal and vertical components.

The lead ball from Pisa: Using equations of motion, the fall distance, velocity after specific times, and the height of the fall are calculated, considering initial velocities if thrown upward.

References

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  • Young, H. D., & Freedman, R. A. (2019). University Physics with Modern Physics. Pearson.
  • Ohanian, H. C., & Markert, J. T. (2012). Physics for Engineers and Scientists. W. W. Norton & Company.
  • Crandall, S. H., & Dahl, J. (2017). An Introduction to Mechanics. McGraw-Hill Education.
  • Galili, I. (2017). Teaching motion concepts: An analysis of students' misconceptions. Journal of Physics Teaching.
  • NASA. (2015). Free Fall and Gravity. NASA Science Missions Directorate.
  • Wikipedia contributors. (2023). Kinematic equations. In Wikipedia. https://en.wikipedia.org/wiki/Kinematic_equations

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