FIGURE EX1.9 shows five points of a motion diagram. Use Tactics Box 1.2 to find the average acceleration vectors at points 1, 2, and 3. Draw the completed motion diagram showing velocity vectors and acceleration vectors.

Verified step by step guidance

1. Step 1: Understand the problem. The motion diagram shows the positions of an object at different times. To find the average acceleration vectors at points 1, 2, and 3, we need to calculate the change in velocity over the change in time between consecutive points. Acceleration is a vector quantity, so direction matters.
2. Step 2: Calculate the average acceleration at point 1. Use the formula: $a=\frac{∆v}{∆}$. Find the velocity vectors at points 1 and 2 by analyzing the displacement between these points and dividing by the time interval. Then, compute the change in velocity ($∆v$) and divide by the time interval ($∆t$).
3. Step 3: Calculate the average acceleration at point 2. Repeat the process for points 2 and 3. Determine the velocity vectors at points 2 and 3, calculate the change in velocity, and divide by the time interval to find the average acceleration vector at point 2.
4. Step 4: Calculate the average acceleration at point 3. Follow the same procedure for points 3 and 4. Determine the velocity vectors at points 3 and 4, calculate the change in velocity, and divide by the time interval to find the average acceleration vector at point 3.
5. Step 5: Draw the completed motion diagram. Represent the velocity vectors at each point as arrows tangent to the path of motion, with their lengths proportional to the magnitude of the velocity. Draw the acceleration vectors at points 1, 2, and 3 as arrows pointing in the direction of the average acceleration, with their lengths proportional to the magnitude of the acceleration. Ensure the diagram clearly shows the relationship between velocity and acceleration vectors.

Key Concepts

Average Acceleration

Average acceleration is defined as the change in velocity over a specified time interval. It is a vector quantity, meaning it has both magnitude and direction. To calculate average acceleration, you subtract the initial velocity from the final velocity and divide by the time taken for that change. This concept is crucial for understanding how an object's speed and direction change over time.

Motion Diagrams

Motion diagrams are visual representations that illustrate an object's position, velocity, and acceleration at various points in time. Each point in the diagram corresponds to a specific moment, showing how the object's motion evolves. By analyzing these diagrams, one can gain insights into the dynamics of the motion, including the relationships between position, velocity, and acceleration.

Vector Representation

Vectors are mathematical entities that have both magnitude and direction, making them essential for representing physical quantities like velocity and acceleration. In motion diagrams, velocity vectors indicate the speed and direction of an object's movement, while acceleration vectors show how the velocity changes. Understanding vector representation is vital for accurately interpreting and drawing motion diagrams, as it allows for a clear depiction of the forces acting on an object.