Math Problem Statement
Productive Task
A particle is moving along a horizontal line in simple harmonic motion. The displacement of the particle at time t is given by:
x(t) = 5 (sin (2t))5
Find the velocity [w(t) = x'(t)] of the
particle at time t =
π /4
Solution
To find the velocity of the particle at time , we first need to differentiate the displacement function to find .
Step 1: Differentiate
Using the chain rule, we differentiate :
Let . Then, . Using the chain rule, we have:
Now, calculate :
Now substituting back:
Thus, the velocity function is:
Step 2: Evaluate at
Now, substitute :
Calculating the sine and cosine:
Thus,
Conclusion
The velocity of the particle at is:
If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:
- How do you derive the displacement function for a different simple harmonic motion?
- What are the characteristics of simple harmonic motion?
- Can you explain the physical significance of velocity being zero in this context?
- How would you find the acceleration of the particle at ?
- What changes if the amplitude or frequency of the motion is altered?
Tip: When dealing with derivatives in physics, always remember to keep track of your units for clarity!
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Simple Harmonic Motion
Formulas
x(t) = 5 (sin(2t))^5
w(t) = x'(t)
Theorems
Chain Rule
Suitable Grade Level
Grades 11-12
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