Math Problem Statement
Solve
Solution
Let's solve each problem one by one from the uploaded image:
Problem 1
Given: We need to find the value of for which the velocity is increasing.
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Find the Velocity : The velocity is the first derivative of with respect to :
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Find the Acceleration : Acceleration is the derivative of with respect to :
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Condition for Velocity to Increase: For the velocity to be increasing, the acceleration must be positive: Solving for : Therefore, the velocity is increasing for .
Problem 2
Given: We need to find the velocity when the acceleration is .
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Find the Velocity : Velocity is the first derivative of with respect to :
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Find the Acceleration : Acceleration is the derivative of with respect to :
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Set Acceleration Equal to 2: We are given that the acceleration is , so: Solving for :
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Find the Velocity at : Substitute into the expression for :
Thus, the velocity is when the acceleration is .
Do you need further details on any of these steps?
Additional Questions:
- What is the physical interpretation of acceleration being positive or negative?
- How does the sign of velocity relate to the direction of motion?
- Could we find the time when the particle stops momentarily using these equations?
- What would happen to the position function if we modified it slightly, like ?
- How can we interpret the derivative of acceleration, and what does it represent physically?
Tip:
When analyzing motion, remember that a positive acceleration does not always mean an increase in speed; it depends on the direction of velocity as well.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Rectilinear Motion
Velocity
Acceleration
Formulas
Velocity v = ds/dt
Acceleration a = dv/dt
Theorems
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Suitable Grade Level
Grades 11-12
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