A rock is dropped from a height of 60 ft . Its height above ground at time \( t \) seconds later is given by \( s(t)=-11 t^{2}+60,0 \leq t \leq 2 \). Find its instantaneous velocity 1 second after it is dropped. The instantaneous velocity is Number \( \mathrm{ft} / \mathrm{sec} \).

To find the instantaneous velocity of the rock 1 second after it is dropped, we can take the derivative of the height function \( s(t) = -11t^2 + 60 \) and evaluate it at \( t = 1 \). The derivative \( s'(t) \) represents the instantaneous velocity: \[ s'(t) = \frac{d}{dt}(-11t^2 + 60) = -22t \] Now, we can substitute \( t = 1 \): \[ s'(1) = -22(1) = -22 \, \text{ft/sec} \] Therefore, the instantaneous velocity of the rock 1 second after it is dropped is \(-22 \, \text{ft/sec}\). --- Did you know that the concept of instantaneous velocity was pivotal to the development of calculus? Sir Isaac Newton and Gottfried Wilhelm Leibniz independently developed the foundational principles that explained motion, allowing us to calculate how fast something is moving at any given moment! In real life, understanding instantaneous velocity is crucial for a variety of fields, especially in engineering and physics. For instance, in designing roller coasters, engineers must account for the instantaneous speeds of the coaster at different points to ensure safety and optimal thrill!