Given the following position function: (position is in ft , time is in sec ) \( \mathrm{s}(\mathrm{t})=7 \mathrm{t}^{2}+\mathrm{t}^{3} \) Find the acceleration at \( \mathrm{t}=3 \). \( \mathrm{a}(3)=[?] \frac{\mathrm{ft}}{\mathrm{sec}^{2}} \)

To find the acceleration at \( t = 3 \) seconds for the given position function \( s(t) = 7t^2 + t^3 \), we first need to compute the velocity function \( v(t) \) by taking the first derivative of the position function with respect to time: \[ v(t) = \frac{ds}{dt} = \frac{d}{dt}(7t^2 + t^3) = 14t + 3t^2 \] Next, we find the acceleration function \( a(t) \) by taking the derivative of the velocity function: \[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(14t + 3t^2) = 14 + 6t \] Now, we substitute \( t = 3 \) into the acceleration function: \[ a(3) = 14 + 6(3) = 14 + 18 = 32 \, \text{ft/sec}^2 \] Thus, the acceleration at \( t = 3 \) is \( \boxed{32} \frac{\text{ft}}{\text{sec}^2} \). In the world of physics, understanding kinematics is crucial. It allows engineers to design everything from roller coasters to vehicles, ensuring they move safely and smoothly. Recognizing the relationships between position, velocity, and acceleration can lead to innovative solutions in motion planning! If you're keen to dive deeper, consider exploring "Calculus-based Physics" textbooks, which often cover these concepts extensively. Additionally, online platforms like Khan Academy have excellent video tutorials that break down kinematic equations, making them easy to grasp and apply to real-life scenarios!