Question

An object is moving with acceleration

a(t)=⟨e⁻²ᵗ ,4t² ,3e⁻³ᵗ ⟩. Find velocity v(t) and position r(t) with the initial conditions r′(0)=⟨0,1,2⟩and r (0)=⟨0,0,0⟩.

Answer 1

To find the velocity and position of an object with a given acceleration, we need to integrate the acceleration function with respect to time. First, let's find the velocity function v(t)...

Step-by-step explanation:

To find the velocity and position of an object with a given acceleration, we need to integrate the acceleration function with respect to time. First, let's find the velocity function v(t):

v(t) = ∫ a(t) dt = ∫ (e-2t, 4t2, 3e-3t) dt

To integrate each component of the acceleration function, we can use the power rule for integration, e.g., ∫ tn dt = (1 / (n+1)) tn+1. After integrating and applying the initial conditions, we can find the position function r(t) by integrating the velocity function:

r(t) = ∫ v(t) dt

Let's calculate the integrals to find v(t) and r(t) with the given initial conditions.