Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


e. When is it moving fastest (highest speed)? Slowest?


s = 4 - 7t + 6t² - t³, 0 ≤ t ≤ 4

Verified step by step guidance

1. Step 1: Begin by identifying the position function given in the problem, which is s(t) = 4 - 7t + 6t² - t³. This function describes the position of the object along the s-axis as a function of time t.
2. Step 2: To find the velocity function v(t), calculate the first derivative of the position function s(t) with respect to time t. This derivative, v(t) = ds/dt = f'(t), represents the rate of change of position, or how fast the object is moving.
3. Step 3: Next, determine the acceleration function a(t) by taking the derivative of the velocity function v(t). This second derivative, a(t) = d²s/dt² = f''(t), indicates the rate of change of velocity, or how the speed of the object is changing over time.
4. Step 4: Graph the position function s(t), the velocity function v(t), and the acceleration function a(t) over the interval 0 ≤ t ≤ 4. This visual representation will help in analyzing the behavior of the object.
5. Step 5: Analyze the graphs to comment on the object's behavior. Look for points where the velocity v(t) is zero, which indicates a change in direction or a momentary stop. The object moves fastest when the magnitude of v(t) is at its maximum. The acceleration a(t) will provide insights into whether the object is speeding up or slowing down. Consider the signs and values of v(t) and a(t) to determine when the object is moving fastest and slowest.

Key Concepts

Position, Velocity, and Acceleration

In calculus, the position function s(t) describes the location of an object over time. The velocity function v(t) is the first derivative of the position function, representing the rate of change of position, or speed and direction. Acceleration a(t) is the second derivative of the position function, indicating the rate of change of velocity. Understanding these relationships is crucial for analyzing motion.

Graphical Analysis of Derivatives

Graphing the position, velocity, and acceleration functions helps visualize an object's motion. The velocity graph shows where the object is moving fastest or slowest, indicated by the peaks and troughs. The acceleration graph reveals changes in velocity, with positive values indicating speeding up and negative values indicating slowing down. Analyzing these graphs provides insights into the object's behavior over time.

Critical Points and Inflection Points

Critical points occur where the derivative (velocity) is zero or undefined, indicating potential maximum or minimum speeds. Inflection points occur where the second derivative (acceleration) changes sign, indicating a change in the concavity of the position graph. Identifying these points helps determine when the object is moving fastest or slowest and provides a deeper understanding of its motion dynamics.