Question

The function s(t) describes the position of a particle moving along a coordinate line, where s is in feet and t is in seconds.

s(t)=t⁴-162 t²+6561, t ≥ 0
Find the velocity and acceleration functions

Answer 1

To find the velocity and acceleration functions for the given position function s(t), we take the first and second derivatives respectively. The velocity function is v(t) = 4t^3 - 324t, and the acceleration function is a(t) = 12t^2 - 324.

Step-by-step explanation:

The question asks to find the velocity and acceleration functions for a particle moving along a coordinate line, given the position function s(t) = t⁴ - 162t² + 6561.

To find the velocity function, we take the first derivative of the position function with respect to time t. The velocity function v(t) is given by:

v(t) = s'(t) = 4t³ - 324t

To find the acceleration function, we take the second derivative of the position function, which is the first derivative of the velocity function. The acceleration function a(t) is given by:

a(t) = v'(t) = 12t² - 324

Both functions describe the rates of change of the particle's position: the velocity function describes the rate of change of position, while the acceleration function describes the rate of change of velocity over time.