The height of a projectile shot from ground level is given...

The height of a projectile shot from ground level is given by $s=-16 t^{2}+256 t,$ where $s$ is measured in feet and $t$ in seconds. (a) Determine the height of the projectile at $t=2, t=6$, $t=9,$ and $t=10$ (b) What is the average velocity of the projectile between $t=2$ and $t=5 ?$ (c) Show that the average velocity between $t=7$ and $t=9$ is zero. Interpret physically. (d) At what time does the projectile hit the ground? (e) Use (8) to find a formula for instantaneous velocity $v$ at a general time $t$. (f) Using the time found in part (d) and the formula found in part (e), find the corresponding impact velocity. (g) What is the maximum height the projectile attains?

The height of a projectile shot from ground level is given by $s=-16 t^{2}+256 t,$ where $s$ is measured in feet and $t$ in seconds.
(a) Determine the height of the projectile at $t=2, t=6$, $t=9,$ and $t=10$
(b) What is the average velocity of the projectile between $t=2$ and $t=5 ?$
(c) Show that the average velocity between $t=7$ and $t=9$ is zero. Interpret physically.
(d) At what time does the projectile hit the ground?
(e) Use (8) to find a formula for instantaneous velocity $v$ at a general time $t$.
(f) Using the time found in part (d) and the formula found in part (e), find the corresponding impact velocity.
(g) What is the maximum height the projectile attains?

Key Concepts

Root Finding in Quadratic Equations

In projectile problems, finding the time when the projectile hits the ground involves solving the quadratic equation obtained by setting the height equal to zero. This process of finding the roots provides the times at which the projectile’s vertical displacement is zero, indicating its launch and landing times.

Maximum Height via Vertex of a Parabola

The maximum height of a projectile is reached at the vertex of the parabolic trajectory, which is the highest point on the graph of the quadratic function. The vertex can be found either by using the formula -b/(2a) for the time coordinate or by completing the square, and it provides critical information regarding the peak altitude and the overall motion of the projectile.

Instantaneous Velocity and Derivatives

Instantaneous velocity is the velocity of the projectile at a specific moment in time, and it can be obtained by taking the derivative of the position function with respect to time. This concept is essential for understanding how the speed of a projectile changes at every point along its trajectory.

Projectile Motion

This concept involves the motion of objects under the influence of gravity, where the only acceleration acting is due to gravity. It includes analyzing the path (trajectory) of an object launched into the air, which is typically represented by quadratic functions in time when air resistance is neglected.

Quadratic Functions in Kinematics

Quadratic functions are commonly used to model the vertical motion of projectiles. They describe how the height of a projectile changes with time when it is subject to a constant acceleration, typically due to gravity. The coefficients in the quadratic equation determine the shape of the trajectory, its vertex, and the times at which the projectile is at specific heights.

Average Velocity

Average velocity in the context of kinematics is defined as the total displacement divided by the time interval over which the displacement occurs. It provides information on the overall change in position over a period of time without revealing the variations in instantaneous speeds during that interval.

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