Given displacement function:

Hey @midnight_puff — ready when you are. To solve this problem, we first need to find the expressions for velocity and acceleration by differentiating the displacement function. Given displacement function: S(t) = t^4 - 2t^2 - 5t^3 We can reorder it for clarity: S(t) = t^4 - 5t^3 - 2t^2 Velocity is the first derivative of displacement with respect to time: V(t) = (dS)/(dt) = (d)/(dt)(t^4 - 5t^3 - 2t^2) = 4t^3 - 15t^2 - 4t Acceleration is the first derivative of velocity with respect to time (or the second derivative of displacement): A(t) = (dV)/(dt) = (d)/(dt)(4t^3 - 15t^2 - 4t) = 12t^2 - 30t - 4 Now we can solve each part of the question. A) Determine when t=3: Step 1: Calculate displacement at t=3. Substitute t=3 into the displacement function S(t). S(3) = (3)^4 - 5(3)^3 - 2(3)^2 S(3) = 81 - 5(27) - 2(9) S(3) = 81 - 135 - 18 S(3) = -72 m The displacement at t=3 is -72 m. Step 2: Calculate velocity at t=3. Substitute t=3 into the velocity function V(t). V(3) = 4(3)^3 - 15(3)^2 - 4(3) V(3) = 4(27) - 15(9) - 12 V(3) = 108 - 135 - 12 V(3) = -39 m/s The velocity at t=3 is -39 m/s. Step 3: Calculate acceleration at t=3. Substitute t=3 into the acceleration function A(t). A(3) = 12(3)^2 - 30(3) - 4 A(3) = 12(9) - 90 - 4 A(3) = 108 - 90 - 4 A(3) = 14 m/s^2 The acceleration at t=3 is 14 m/s^2. B) Time when velocity is 0: Step 4: Set the velocity function V(t) to 0 and solve for t. V(t) = 4t^3 - 15t^2 - 4t = 0 Factor out t: t(4t^2 - 15t - 4) = 0 This gives two possibilities: t=0 or 4t^2 - 15t - 4 = 0. Solve the quadratic equation 4t^2 - 15t - 4 = 0. We can factor it: (4t+1)(t-4) = 0 This yields two more solutions: 4t+1 = 0 t = -(1)/(4) s t-4 = 0 t = 4 s Since time cannot be negative in this physical context, the times when velocity is 0 are t=0 s and t=4 s. The times when velocity is 0 are 0 s and 4 s. C) Acceleration at t=2: Step 5: Calculate acceleration at t=2. Substitute t=2 into the acceleration function A(t). A(2) = 12(2)^2 - 30(2) - 4 A(2) = 12(4) - 60 - 4 A(2) = 48 - 60 - 4 A(2) = -16 m/s^2 The acceleration at t=2 is -16 m/s^2. Drop the next question!