Impact speed A rock is dropped off the edge of a cliff and...

Impact speed A rock is dropped off the edge of a cliff and its distance $s$ (in feet) from the top of the cliff after $t$ seconds is $s(t)=16 t^{2} .$ Assume the distance from the top of the cliff to the ground is $96 \mathrm{ft}.$ a. When will the rock strike the ground? b. Make a table of average velocities and approximate the velocity at which the rock strikes the ground.

Impact speed A rock is dropped off the edge of a cliff and its distance $s$ (in feet) from the top of the cliff after $t$ seconds is $s(t)=16 t^{2} .$ Assume the distance from the top of the cliff to the ground is $96 \mathrm{ft}.$
a. When will the rock strike the ground?
b. Make a table of average velocities and approximate the velocity at which the rock strikes the ground.

Key Concepts

Graphing Functions

Graphing functions involves plotting the points of a function in a coordinate system to visualize its overall behavior, including key features like intercepts, turning points, and symmetry. This process is essential for understanding and analyzing the shape and properties of the function.

Parabolic Functions

Parabolic functions are quadratic functions whose graphs form a parabola. Key characteristics of a parabola include its vertex, which is the maximum or minimum point, its axis of symmetry, and its opening direction (upward or downward). These features are foundational in studying the behavior of quadratic relations.

Tangent Lines and Zero Slope

A tangent line to a function at a given point is a line that touches the curve at that point and has the same instantaneous rate of change as the function. A tangent line with zero slope, meaning it is horizontal, often indicates a local extremum, such as a maximum or minimum, where the derivative of the function is zero.

Secant Lines

A secant line intersects the graph of a function at two distinct points and represents the average rate of change between those points. The concept of secant lines helps in understanding the transition from average rates of change to instantaneous rates of change (derivatives) as the two points become arbitrarily close.

Symmetry

Symmetry in a function, particularly in quadratic functions, refers to the mirror-image property of the graph about a central axis. This concept simplifies the analysis of the function's properties, such as the behavior of slopes of secant and tangent lines, by providing predictable structural behavior in the graph.

Transcript

00:01 In this question we have got the position of rock falling from cliff has a function of time and that is given as position.

00:12 So that is s of t as equals to 16 t square.

00:18 Okay.

00:19 So the first question they asked when the rock will hit the ground, okay? and the height of the cliff is, height of the cliff is 96 feet.

00:37 So for the first question we have to find the time when this rock which is falling from the cliff will hit the ground.

00:46 So essentially for that what we will be doing is the position function is given as s of t.

00:51 So we will equate the position function with height of the cliff and we'll get the time.

00:57 So in this case 96 will be equals to 16 t square.

01:03 Okay, 16, 6xia, 96 and then.

01:08 And if we calculate it to t we are getting is root of 6 seconds.

01:14 Okay, so the rock will hit the ground at the time t is equal to root 6 seconds and that is answer for the part a.

01:23 In part b, we have to calculate the speed or the velocity of the rock when it hits the ground.

01:34 So, for that what we will be doing is we will differentiate this function of position.

01:39 So t squared, so velocity will be equals to or d s by d t.

01:46 So what will be getting? we will be getting 32t.

01:51 Okay? and the time we already have, time is root 6...

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