As a particle moves along the number line, its position at time t is s(t), its velocity is v(t), and its acceleration is a(t)=3t^(2). If v(0)=3 and s(0)=1, what is s(2)?

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As a particle moves along the number line, its position at time  t is  s(t), its velocity is  v(t), and its acceleration is  a(t)=3t^(2). If  v(0)=3 and  s(0)=1, what is  s(2)?

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Given Data: We are given the acceleration function a(t) = 3t^2, the initial velocity v(0) = 3, and the initial position s(0) = 1. To find the position s(2), we first need to find the velocity function v(t) by integrating the acceleration function a(t).Find Velocity Function: Integrate the acceleration function a(t) = 3t^2 to find the velocity function v(t). The indefinite integral of 3t^2 with respect to t is t^3, so v(t) = t^3 + C, where C is the constant of integration.Constant of Integration: To find the constant of integration C, we use the initial condition v(0) = 3. Plugging t = 0 into the velocity function gives us 0^3 + C = 3. Therefore, C = 3 and the velocity function is v(t) = t^3 + 3.Find Position Function: Next, we integrate the velocity function v(t) = t^3 + 3 to find the position function s(t). The indefinite integral of t^3 is (1/4)t^4, and the indefinite integral of 3 is 3t. So, s(t) = (1/4)t^4 + 3t + D, where D is another constant of integration.Constant of Integration: To find the constant of integration D, we use the initial condition s(0) = 1. Plugging t = 0 into the position function gives us (1/4)*0^4 + 3*0 + D = 1. Therefore, D = 1 and the position function is s(t) = (1/4)t^4 + 3t + 1.Evaluate Position Function: Finally, we evaluate the position function s(t) at t = 2 to find s(2). Plugging t = 2 into s(t) gives us s(2) = (1/4)*(2)^4 + 3*(2) + 1.Calculate Final Position: Calculating s(2), we have s(2) = (1/4)*16 + 3*2 + 1 = 4 + 6 + 1 = 11.

As a particle moves along the number line, its position at time $t$ is $s(t)$ , its velocity is $v(t)$ , and its acceleration is $a(t)=3 t^{2}$ . $\newline$ If $v(0)=3$ and $s(0)=1$ , what is $s(2) ?$

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As a particle moves along the number line, its position at time t t t is s(t) s(t) s(t), its velocity is v(t) v(t) v(t), and its acceleration is a(t)=3t2 a(t)=3 t^{2} a(t)=3t2.\newlineIf v(0)=3 v(0)=3 v(0)=3 and s(0)=1 s(0)=1 s(0)=1, what is s(2)? s(2) ? s(2)?

As a particle moves along the number line, its position at time $t$ is $s(t)$ , its velocity is $v(t)$ , and its acceleration is $a(t)=3 t^{2}$ . $\newline$ If $v(0)=3$ and $s(0)=1$ , what is $s(2) ?$