Question

A particle moves in a straight line and has acceleration given by a(t) =6t+4. Its initial velocity is v(o) =-6m/s and its initial displacement s(0) =9m. What is its position function s(t)?

Answer 1

To find the position function s(t) of the particle, we need to integrate the acceleration function twice with respect to time.

Given: Acceleration function: a(t) = 6t + 4 Initial velocity: v(0) = -6 m/s Initial displacement: s(0) = 9 m

First, let's integrate the acceleration function to find the velocity function v(t): v(t) = ∫(6t + 4) dt v(t) = 3t^2 + 4t + C1

Using the initial velocity condition, v(0) = -6 m/s: -6 = 3(0)^2 + 4(0) + C1 -6 = C1

So the velocity function becomes: v(t) = 3t^2 + 4t - 6

Now, let's integrate the velocity function to find the position function s(t): s(t) = ∫(3t^2 + 4t - 6) dt s(t) = t^3 + 2t^2 - 6t + C2

Using the initial displacement condition, s(0) = 9 m: 9 = (0)^3 + 2(0)^2 - 6(0) + C2 9 = C2

So the position function becomes: s(t) = t^3 + 2t^2 - 6t + 9

Therefore, the position function of the particle is s(t) = t^3 + 2t^2 - 6t + 9.