Answer:
x(t=3s) = 0.07 m to the nearest hundredth
Explanation:
v(t) = t² e⁻³ᵗ
Find displacement after t = 3 s.
Recall, velocity, v = (dx/dt)
v = (dx/dt) = t² e⁻³ᵗ
dx = t² e⁻³ᵗ dt
∫ dx = ∫ t² e⁻³ᵗ dt
This integration will be done using the integration by parts method.
Integration by parts is done this way...
∫ u dv = uv - ∫ v du
Comparing ∫ t² e⁻³ᵗ dt to ∫ u dv
u = t²
∫ dv = ∫ e⁻³ᵗ dt
u = t²
(du/dt) = 2t
du = 2t dt
∫ dv = ∫ e⁻³ᵗ dt
v = (-e⁻³ᵗ/3)
∫ u dv = uv - ∫ v du
Substituting the variables for u, v, du and dv
∫ t² e⁻³ᵗ dt = (-t²e⁻³ᵗ/3) - ∫ (-e⁻³ᵗ/3) 2t dt
= (-t²e⁻³ᵗ/3) - ∫ 2t (-e⁻³ᵗ/3) dt
But the integral (∫ 2t (-e⁻³ᵗ/3) dt) is another integration by parts problem.
∫ u dv = uv - ∫ v du
u = 2t
∫ dv = ∫ (-e⁻³ᵗ/3) dt
u = 2t
(du/dt) = 2
du = 2 dt
∫ dv = ∫ (-e⁻³ᵗ/3) dt
v = (e⁻³ᵗ/9)
∫ u dv = uv - ∫ v du
Substituting the variables for u, v, du and dv
∫ 2t (-e⁻³ᵗ/3) dt = 2t (e⁻³ᵗ/9) - ∫ 2 (e⁻³ᵗ/9) dt = 2t (e⁻³ᵗ/9) + (2e⁻³ᵗ/27)
Putting this back into the main integration by parts equation
∫ t² e⁻³ᵗ dt = (-t²e⁻³ᵗ/3) - ∫ 2t (-e⁻³ᵗ/3) dt = (-t²e⁻³ᵗ/3) - [2t (e⁻³ᵗ/9) + (2e⁻³ᵗ/27)]
x(t) = ∫ t² e⁻³ᵗ dt = (-t²e⁻³ᵗ/3) - 2t (e⁻³ᵗ/9) - (2e⁻³ᵗ/27) + k (k = constant of integration)
x(t) = (-t²e⁻³ᵗ/3) - 2t (e⁻³ᵗ/9) - (2e⁻³ᵗ/27) + k
At t = 0 s, v(0) = 0, hence, x(0) = 0
0 = 0 - 0 - (2/27) + k
k = (2/27)
x(t) = (-t²e⁻³ᵗ/3) - 2t (e⁻³ᵗ/9) - (2e⁻³ᵗ/27) + (2/27)
At t = 3 s
x(3) = (-9e⁻⁹/3) - (6e⁻⁹/9) - (2e⁻⁹/27) + (2/27)
x(3) = -0.0003702294 - 0.0000822732 - 0.0000091415 + 0.0740740741 = 0.07361243 m = 0.07 m to the nearest hundredth.
