Math Problem Statement
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds).
s (m)
What is the body's velocity when t = 2.5 sec?
Solution
The image shows a graph of position (in meters) as a function of time (in seconds). To find the velocity of the body at seconds, we need to determine the slope of the position function at that specific time. The velocity is the rate of change of position with respect to time, which is the derivative of the position function .
Step-by-step process:
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Locate on the graph: From the graph, lies between and . We need to find the slope of the graph in this interval.
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Analyze the slope in the interval :
- The graph shows a flat line between and . This indicates that the position of the body does not change, meaning that the velocity is during this time interval.
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Conclusion: Since the slope of the position graph is zero at , the velocity of the body at seconds is meters per second.
Let me know if you'd like further details or explanations!
Here are five related questions for deeper exploration:
- What is the body's acceleration at seconds?
- How would you calculate the velocity at seconds?
- What is the body's total displacement from to ?
- How would the velocity change if the slope of the graph between and were steeper?
- How can you use the graph to find when the body changes direction?
Tip: Always check if the graph is increasing, decreasing, or flat to determine the sign and magnitude of velocity.
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Math Problem Analysis
Mathematical Concepts
Position Function
Velocity
Rate of Change
Slope of a Function
Formulas
Velocity = Derivative of Position Function (ds/dt)
Slope = Δy/Δx
Theorems
Derivative as Rate of Change
Graphical Interpretation of Derivatives
Suitable Grade Level
Grades 9-12
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