Math Problem Statement
An object moves along the x axis according to the equation x = 4.00t2 − 2.00t + 3.00, where x is in meters and t is in seconds. (a) Determine the average speed between t = 1.50 s and t = 3.40 s. m/s
(b) Determine the instantaneous speed at t = 1.50 s. m/s
Determine the instantaneous speed at t = 3.40 s. m/s
(c) Determine the average acceleration between t = 1.50 s and t = 3.40 s. m/s2
(d) Determine the instantaneous acceleration at t = 1.50 s. m/s2
Determine the instantaneous acceleration at t = 3.40 s. m/s2
(e) At what time is the object at rest?
Solution
Given the equation of motion for an object moving along the x-axis: where is in meters and is in seconds, we'll determine several key parameters:
(a) Average Speed between and
The average speed is calculated as: where and .
First, find the positions at and : Now, compute the average speed.
(b) Instantaneous Speed at and
To find the instantaneous speed, we first need to calculate the velocity by taking the derivative of the position function: At : At : Since speed is the magnitude of velocity, the instantaneous speeds will be the absolute values of the velocities at these times.
(c) Average Acceleration between and
The average acceleration is calculated as: Substitute the values of and that were calculated earlier.
(d) Instantaneous Acceleration at and
To find the instantaneous acceleration, we take the derivative of the velocity function: Thus, the instantaneous acceleration is constant and equal to at both and .
(e) Time when the object is at rest
The object is at rest when the velocity is zero. Set the velocity equation to zero and solve for :
Now, let's calculate these values step by step.Here are the results for the problem:
(a) Average speed between and :
(b) Instantaneous speed at:
- :
- :
(c) Average acceleration between and :
(d) Instantaneous acceleration (constant) at:
- :
- :
(e) The object is at rest at .
Would you like further details or clarification on any part of this?
Here are five related questions:
- How do you calculate average velocity if the motion is non-uniform?
- What is the difference between speed and velocity?
- How does acceleration change in uniformly accelerated motion?
- What physical interpretation can be given to negative acceleration?
- How do you find displacement from a velocity-time graph?
Tip: Instantaneous speed is always the magnitude of the velocity, while average speed is the total distance divided by total time.
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Math Problem Analysis
Mathematical Concepts
Kinematics
Average Speed
Instantaneous Speed
Average Acceleration
Instantaneous Acceleration
Formulas
Average speed = |x(t2) - x(t1)| / (t2 - t1)
Instantaneous speed = |v(t)|
Velocity v(t) = dx/dt
Average acceleration = (v(t2) - v(t1)) / (t2 - t1)
Instantaneous acceleration a(t) = dv/dt
Theorems
First derivative of position gives velocity
Second derivative of position gives acceleration
Suitable Grade Level
Grades 10-12
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