Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

The position of a particle is given by the function [tex]\( x = 2t^3 - 8t^2 + 12 \)[/tex] m, where [tex]\( t \)[/tex] is in seconds.

Part A:
At what time does the particle reach its minimum velocity?


Sagot :

To determine when the particle reaches its minimum velocity, we can follow these steps:

1. Define the position function:
The position of the particle is given by:
[tex]\[ x(t) = 2t^3 - 8t^2 + 12 \][/tex]

2. Find the velocity function:
The velocity is the first derivative of the position function with respect to time [tex]\( t \)[/tex]:
[tex]\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(2t^3 - 8t^2 + 12) \][/tex]
Calculating this derivative, we get:
[tex]\[ v(t) = 6t^2 - 16t \][/tex]

3. Find the acceleration function:
The acceleration is the first derivative of the velocity function with respect to time [tex]\( t \)[/tex]:
[tex]\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(6t^2 - 16t) \][/tex]
Calculating this derivative, we get:
[tex]\[ a(t) = 12t - 16 \][/tex]

4. Determine when the acceleration is zero:
To find the time when the velocity is at a minimum, we set the acceleration function to zero and solve for [tex]\( t \)[/tex]:
[tex]\[ 12t - 16 = 0 \][/tex]
Solving for [tex]\( t \)[/tex], we get:
[tex]\[ t = \frac{16}{12} = \frac{4}{3} \][/tex]

5. Verify that the velocity at this time is a minimum:
We use the second derivative test to confirm that we have a minimum. The second derivative of acceleration (or the third derivative of the position function) is:
[tex]\[ a''(t) = \frac{d^2v}{dt^2} = \frac{d}{dt}(12t - 16) = 12 \][/tex]
Since [tex]\( a''(t) = 12 \)[/tex] is a positive constant, [tex]\( t = \frac{4}{3} \)[/tex] is indeed a minimum.

Thus, the particle reaches its minimum velocity at [tex]\( t = \frac{4}{3} \)[/tex] seconds.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.