Final answer:
To find the time when the particle has a velocity of 20 m/s, we need to find the time when the velocity function is equal to 20 m/s. The velocity function is the derivative of the position function. By taking the derivative and solving the resulting equation, we find that the particle has a velocity of 20 m/s at t = 5 seconds.
Step-by-step explanation:
To find the time when the particle has a velocity of 20 m/s, we need to find the time when the velocity function is equal to 20 m/s. In this case, the velocity function is the derivative of the position function. So, we need to find the derivative of the position function S(t) with respect to time and equate it to 20.
Given: S(t) = t⁴ - 4t³ - 20t² + 20t
Taking the derivative, v(t) = dS(t)/dt = 4t³ - 12t² - 40t + 20
Setting v(t) equal to 20 and solving for t, we have: 4t³ - 12t² - 40t + 20 = 20
Simplifying the equation, 4t³ - 12t² - 40t + 20 - 20 = 0
Simplifying further, 4t³ - 12t² - 40t = 0
Factoring out a common factor of 4t, we get: 4t(t² - 3t - 10) = 0
Setting each factor equal to zero, we find two possible values for t: t = 0 and t² - 3t - 10 = 0
Solving the quadratic equation t² - 3t - 10 = 0, we get two solutions for t: t = -2 and t = 5
Since t represents time and time cannot be negative, we can conclude that the particle has a velocity of 20 m/s at t = 5 seconds.