Question

Find the position vector of a particle that has the given acceleration and the specified initial velocity and position.

a(t) = 19t i + e⁻ j + e⁻ᵗ k, v(0) = k, r(0) = j + k

Answer 1

To find the position vector of a particle with the given acceleration, initial velocity, and initial position, integrate the acceleration twice. The position vector is given by (19/3)t³ i - e⁻ᵀ j + e⁻ᵗ k + kt + (j + k).

Step-by-step explanation:

To find the position vector of a particle with the given acceleration, initial velocity, and initial position, we can integrate the acceleration function twice.

1. Integrate the acceleration function to find the velocity function:

v(t) = ∫(a(t)dt) = ∫(19t i + e⁻ j + e⁻ᵗ k)dt = 19t² i - e⁻ j - e⁻ᵗ k + C₁

1. Integrate the velocity function to find the position function:

r(t) = ∫(v(t)dt) = ∫((19t² i - e⁻ j - e⁻ᵗ k + C₁)dt) = (19/3)t³ i - e⁻ᵀ j + e⁻ᵗ k + C₁t + C₂

To find the constants C₁ and C₂, we use the initial velocity and position:

v(0) = k = 0 + C₁ => C₁ = k

r(0) = j + k = 0 + 0 + C₂ => C₂ = j + k

Substituting the values of the constants, we get:

r(t) = (19/3)t³ i - e⁻ᵀ j + e⁻ᵗ k + kt + (j + k)