Math Problem Statement
A particle is moving with a displacement of x(t)=π‘3+3π‘2+1 measured in meters. a) Find the average velocity from t=1s to t=3s b) Find an equation for the instantaneous velocity c) Find the instantaneous velocity at t=1s and simplify your answer.
Solution
We are given the displacement function of a particle, , where is measured in meters and is in seconds. Let's solve each part step by step.
a) Find the average velocity from to
The formula for the average velocity over an interval is: In this case, and . First, let's calculate and .
Now, calculate the average velocity:
Thus, the average velocity from to is .
b) Find an equation for the instantaneous velocity
The instantaneous velocity is the derivative of the displacement function with respect to time . Let's find the derivative of .
Thus, the equation for the instantaneous velocity is:
c) Find the instantaneous velocity at and simplify your answer
To find the instantaneous velocity at , substitute into the equation for :
Thus, the instantaneous velocity at is .
Summary of Results:
a) The average velocity from to is .
b) The equation for the instantaneous velocity is .
c) The instantaneous velocity at is .
Would you like further details or have any questions about these solutions?
Here are 5 additional related questions:
- What would the instantaneous velocity be at ?
- How would you find the acceleration of the particle at any time ?
- How does the velocity change between and ?
- Can you find the time when the particle's velocity is zero?
- What is the significance of the constant term in the displacement function?
Tip: The derivative of a displacement function gives you the velocity, and the derivative of velocity gives you the acceleration!
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Math Problem Analysis
Mathematical Concepts
Calculus
Kinematics
Differentiation
Formulas
Average velocity: v_avg = (x(t2) - x(t1)) / (t2 - t1)
Instantaneous velocity: v(t) = dx/dt
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12
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