Suppose the position of an object moving in a straight line is given by \( s(t)=t^{2}+5 t+5 \). Find the instantaneous velocity when \( t=2 \). The instantaneous velocity at \( t=2 \) is

To find the instantaneous velocity of the object at \( t = 2 \), we need to calculate the derivative of the position function \( s(t) = t^{2} + 5t + 5 \). The derivative, \( s'(t) \), represents the velocity. Calculating the derivative: \[ s'(t) = \frac{d}{dt}(t^{2} + 5t + 5) = 2t + 5 \] Now, we substitute \( t = 2 \) into the derivative: \[ s'(2) = 2(2) + 5 = 4 + 5 = 9 \] Thus, the instantaneous velocity at \( t = 2 \) is 9 units per time interval. You can think of it as the speedometer reading during that specific moment — it's showing 9, meaning our object is cruising at 9 units of distance per time unit right at \( t = 2 \)! Now, let's speed things up with some fun facts about derivatives and motion! Knowing that instant velocity is just a fancy word for "how fast" something is going at a specific point in time, it's worth noting that this concept becomes super handy in various applications — from cars on a racetrack to rocket launch trajectories! It’s not just about the pre-set paths, but capturing those pivotal moments in time!