What is the velocity vector and position vector?

Question

GIven an acceleration vector, initial velocity (u₀, v₀), and initial position (x₀, y₀), find the velocity and position vectors for t>=0.

a(t)=<cos t, 2 sin t>

<u₀, v₀>=<0, 4>

<x₀, y₀> = <4, 0>

Answer 1

Integrate the components of a(t) to get v(t):

v(t) = < sint + c₁, -2cost + c₂ >

Since v(0) = <u₀, v₀>, sin0 + c₁ = u₀ and -2cos0 + c₂ = v_0.

So, c₁ = u₀ and c₂ = v₀ + 2

Therefore, v(t) = < sint + u₀,_, -2cost + 2 + v₀ >

Integrate v(t) componentwise to get s(t), the position vector:

s(t) = < -cost + u₀t + c₃, -2sint + 2t + v₀t + c₄ >

Since s(0) = <x₀, y₀ >, -cos0 + c₃ = x₀ and -2sin0 + c₄ = y₀

So, c₃ = x₀ + 1 and c₄ = y₀

s(t) = < -cost + u₀t + x₀ + 1, -2sint + 2t + v₀t + y₀ >