Math Problem Statement
Solution
The problem describes the motion of a weight attached to a spring, where the displacement is given by:
This is a simple harmonic motion, and the velocity function can be derived by differentiating the displacement with respect to time .
(a) Finding the spring's velocity when and
1. Differentiate the displacement function:
The velocity is the derivative of with respect to :
2. (b) Velocity at :
Substitute into the velocity function: This confirms the answer provided in the image, which is .
3. Velocity at :
Substitute into the velocity function:
Since , we get: Numerically, this is approximately:
Therefore, the velocity at is approximately .
Would you like more details on any specific part of the solution?
Here are 5 follow-up questions to consider:
- What is the maximum velocity of the spring's motion?
- How would the velocity change if the displacement function were ?
- What is the period of this spring's motion?
- How does changing the amplitude affect the velocity at different times?
- What is the spring's velocity at ?
Tip: Remember, velocity is always the derivative of the displacement function when dealing with harmonic motion.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Calculus
Harmonic Motion
Formulas
Displacement function x(t) = 14 cos(t)
Velocity function v(t) = dx/dt = -14 sin(t)
Theorems
Derivative of trigonometric functions
Simple Harmonic Motion
Suitable Grade Level
Grades 11-12
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