A particle is moving with the given data. Find the position of the particle.

Question

a(t) = t² − 3t + 4,    s(0) = 0,    s(1) = 20

Answer 1

In order to determine the position of the particle, we have to take the integral of the acceleration to obtain velocity and then integrate velocity to obtain position:

∫a(t)dt = v(t)

∫v(t)dt = s(t)

v(t) = ∫(t² − 3t + 4)dt = t³/3 - 3t²/2 + 4t + C₁ where C₁ is some constant

s(t) = ∫v(t) = ∫(t³/3 - 3t²/2 + 4t + C₁)dt = t⁴/12 - 3t³/6 + 4t²/2 + C₁t + C₂ where C₂ is some constant

Simplifying, we get:

s(t) = t⁴/12 - t³/2 + 2t² + C₁t + C₂

Now we have to solve for C₁ and C₂ with our initial givens:

s(0) = 0

s(1) = 20

Let's solve for s(0):

s(0) = 0⁴/12 - 0³/2 + 2(0)² + C₁(0) + C₂ = 0

C₂ = 0

Now we can solve s(1) = 20

s(1) = 1⁴/12 - 1³/2 + 2(1)² + C₁(1)

s(1) = 1/12 - 1/2 + 2 + C₁ = 20

s(1) = 19/12 + C₁ = 20

C₁ = 20 - 19/12

C₁ = 221/12

s(t) = t⁴/12 - t³/2 + 2t² + (221/12)t