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

Question

a(t) = 2t + 3,    s(0) = 6,    v(0) = −8

s(t)=



Answer 1

First, it is important to note that acceleration is the derivative of velocity, and velocity is the derivative of position. As such, we can integrate acceleration once to find the velocity, and integrate velocity once to find position. Let's do that now:

a(t) = 2t + 3

v(t) = t² + 3t + C₁

s(t) = (1/3)t³ + (3/2)t² + C₁t + C₂

Now that we have the general form of the equation for position, we need to determine the values for the constants C₁ and C₂. Let's start with v(0) = -8. Plugging 0 into v(t) gives:

v(0) = (0)² + 3(0) + C₁ = -8 or C₁ = -8

Now, we can write s(t) as s(t) = (1/3)t³ + (3/2)t² -8t + C₂. We also know s(0) = 6, so plugging 0 into s(t) gives:

s(0) = (1/3)(0)³ + (3/2)(0)² -8(0) + C₂ = 6 or C₂ = 6.

Thus, our final answer is s(t) = (1/3)t³ + (3/2)t² -8t + 6.