Velocity from position The graph of $s=f\left(t\right)$ represents the position of an object moving along a line at time $t\ge 0$ . <IMAGE>
a. Assume the velocity of the object is 0 when $t=0$ . For what other values of t is the velocity of the object zero?

Verified step by step guidance

1. Step 1: Understand that the velocity of an object is the derivative of its position function with respect to time, i.e., v(t) = f'(t).
2. Step 2: Since the velocity is zero at t = 0, we need to find other values of t where the derivative f'(t) = 0.
3. Step 3: Analyze the graph of s = f(t) to identify points where the tangent to the curve is horizontal, as these correspond to points where f'(t) = 0.
4. Step 4: Look for critical points on the graph where the slope of the tangent line is zero, indicating potential values of t where the velocity is zero.
5. Step 5: Verify these points by considering the behavior of the graph around these critical points to ensure they are indeed points where the velocity is zero.

Key Concepts

Position Function

The position function, denoted as s = f(t), describes the location of an object along a line at any given time t. Understanding this function is crucial because it provides the basis for analyzing the object's motion. The graph of this function visually represents how the position changes over time, allowing us to infer other properties like velocity and acceleration.

Velocity

Velocity is defined as the rate of change of position with respect to time, mathematically expressed as v(t) = f'(t), where f'(t) is the derivative of the position function. It indicates how fast and in what direction the object is moving. When the velocity is zero, it signifies that the object is momentarily at rest, which is essential for determining points of interest in motion analysis.

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. In the context of velocity, finding critical points helps identify when the object's velocity is zero, which corresponds to moments when the object stops or changes direction. Analyzing these points is vital for understanding the overall motion of the object.