Math Problem Statement

The motion of a material point is given by the function y = S'(t) on the interval [0, 8]. The graph of S'(t), which represents the velocity of the point, is shown. Determine the moment in time when the velocity of the point is equal to 3.

Solution

The problem is as follows:

The motion of a material point is given by the function y=S(t)y = S'(t) on the interval [0,8][0, 8]. The graph of S(t)S'(t), which represents the velocity of the point, is shown in Figure 279. You are tasked to determine the moment in time when the velocity of the point is equal to 33.


Solution:

  1. The graph y=S(t)y = S'(t) represents the velocity of the point. From the graph in Figure 279, we need to find the time tt where S(t)=3S'(t) = 3.
  2. Observing the graph:
    • At t=3t = 3, the velocity S(t)=3S'(t) = 3.
    • This is the only time on the graph where S(t)=3S'(t) = 3.

Answer:

The moment in time when the velocity of the point is equal to 33 is t=3t = 3 seconds.


Would you like a detailed explanation of the graph analysis or additional related concepts?

Here are five follow-up questions to deepen your understanding:

  1. What is the significance of the derivative S(t)S'(t) in motion-related problems?
  2. How can you determine acceleration using this graph?
  3. What does the area under the S(t)S'(t) curve represent in terms of the original function S(t)S(t)?
  4. How would you find when the velocity is negative using this graph?
  5. How does this graph relate to real-world motion problems?

Tip: Always identify key values (like maxima, minima, or specific points) directly on a graph to quickly locate critical information.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Derivative
Velocity in motion
Graph interpretation

Formulas

S'(t): The derivative of displacement function represents velocity

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 9-12