Question

suppose that the position of a particle is given by \\(s = f(t) = 2t^3 + 4t + 9\\).

(a) find the velocity at time \\(t\\).
\\(v(t) = \\) \\(\frac{m}{s}\\)

(b) find the velocity at time \\(t = 3\\) seconds.
\\(\frac{m}{s}\\)

(c) find the acceleration at time \\(t\\).
\\(a(t) = \\) \\(\frac{m}{s^2}\\)

(d) find the acceleration at time \\(t = 3\\) seconds.
\\(\frac{m}{s^2}\\)

Explanation:

Find the velocity function

$$ v(t) = s'(t) = \frac{d}{dt}(2t^3 + 4t + 9) = 6t^2 + 4 $$

Evaluate velocity at t = 3

$$ v(3) = 6(3)^2 + 4 = 6(9) + 4 = 58 $$

Find the acceleration function

$$ a(t) = v'(t) = \frac{d}{dt}(6t^2 + 4) = 12t $$

Evaluate acceleration at t = 3

$$ a(3) = 12(3) = 36 $$

Answer:

Question a

Suppose that the position of a particle is given by \(s = f(t) = 2t^3 + 4t + 9\).
(a) Find the velocity at time \(t\).
\(v(t) =\) <blank>\(6t^2 + 4\)</blank> \(\frac{\text{m}}{\text{s}}\)

Question b

(b) Find the velocity at time \(t = 3\) seconds.
<blank>\(58\)</blank> \(\frac{\text{m}}{\text{s}}\)

Question c

(c) Find the acceleration at time \(t\).
\(a(t) =\) <blank>\(12t\)</blank> \(\frac{\text{m}}{\text{s}^2}\)

Question d

(d) Find the acceleration at time \(t = 3\) seconds.
<blank>\(36\)</blank> \(\frac{\text{m}}{\text{s}^2}\)