The altitude (in feet) of a model rocket t sec into a trial...

The altitude (in feet) of a model rocket $t$ sec into a trial flight is given by $$ s=f(t)=-2 t^{3}+12 t^{2}+5 \quad(t \geq 0) $$ a. Find an expression $v$ for the rocket's velocity at any time $t$ b. Compute the rocket's vertical velocity when $t=0,2,4,$ and $6 .$ Interpret your results. c. Using the results from the solution to part (b), find the maximum altitude attained by the rocket. Hint: At its highest point, the velocity of the rocket is zero.

The altitude (in feet) of a model rocket $t$ sec into a trial flight is given by
$$
s=f(t)=-2 t^{3}+12 t^{2}+5 \quad(t \geq 0)
$$
a. Find an expression $v$ for the rocket's velocity at any time $t$
b. Compute the rocket's vertical velocity when $t=0,2,4,$ and $6 .$ Interpret your results.
c. Using the results from the solution to part (b), find the maximum altitude attained by the rocket.
Hint: At its highest point, the velocity of the rocket is zero.

Key Concepts

Rate of Change and Derivatives

A derivative represents the rate at which a quantity changes with respect to another quantity. In the context of motion, the derivative of a position or altitude function with respect to time gives the velocity. This concept is fundamental in calculus for understanding how dynamic systems evolve over time.

Differentiation of Polynomial Functions

Differentiating polynomial functions involves applying rules such as the power rule and constant multiple rule to each term. This process simplifies the function, allowing for the determination of its instantaneous rate of change, which is essential for analyzing physical situations like motion.

Critical Points and Optimization

Critical points occur when the derivative of a function equals zero or is undefined, indicating potential local maxima or minima. In many problems, such as optimizing the height of a projectile, identifying these points helps determine when a function reaches its highest or lowest value.

Physical Interpretation of Derivative Values

In the study of motion, the sign and magnitude of the derivative (velocity) provide important information about the behavior of the moving object. A positive velocity indicates upward motion, while a negative velocity shows downward motion. A velocity of zero typically signifies a turning point, such as the maximum height of a projectile.