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

Question

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

a(t) = 2t + 3,    s(0) = 3,    v(0) = −2

Answer 1

Remember that the derivative of a position function s(t) is the velocity function v(t), and the derivative of the velocity function v(t) is the acceleration function a(t). With this in mind, we can work backwards by taking anti-derivatives of a(t) and v(t) to find the position function.

a(t) = 2t + 3

v(t) = t² + 3t + C₁ [anti derivative]

v(0) = 0 + 0 + C₁ = -2 [using the value of v(0) = -2]

Therefore C₁ = -2 and v(t) = t² + 3t - 2

Now we can do the same process once more to find the position function,

v(t) = t² + 3t - 2

s(t) = (1/3)t³ + (3/2)t² - 2t + C₂ [anti derivative]

s(0) = 0 + 0 - 0 + C₂ = 3

Therefore C₂ = 3 and s(t) = (1/3)t³ + (3/2)t² - 2t + 3