Zero velocity A projectile is fired vertically upward and...

Zero velocity A projectile is fired vertically upward and has a position given by $s(t)=-16 t^{2}+128 t+192,$ for $0 \leq t \leq 9$ a. Graph the position function, for $0 \leq t \leq 9.$ b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time $t=a.$ c. Confirm your answer to part (b) by making a table of average velocities to approximate the instantaneous velocity at $t=a.$ d. For what values of $t$ on the interval [0.9] is the instantaneous velocity positive (the projectile moves upward)? e. For what values of $t$ on the interval [0,9] is the instantaneous velocity negative (the projectile moves downward)?

Zero velocity A projectile is fired vertically upward and has a position given by $s(t)=-16 t^{2}+128 t+192,$ for $0 \leq t \leq 9$
a. Graph the position function, for $0 \leq t \leq 9.$
b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time $t=a.$
c. Confirm your answer to part (b) by making a table of average velocities to approximate the instantaneous velocity at $t=a.$
d. For what values of $t$ on the interval [0.9] is the instantaneous velocity positive (the projectile moves upward)?
e. For what values of $t$ on the interval [0,9] is the instantaneous velocity negative (the projectile moves downward)?

Key Concepts

Projectile Motion

Projectile motion is the study of objects that are launched into the air and influenced solely by gravity (neglecting air resistance). This concept involves analyzing how objects move along a curved trajectory and includes understanding parameters such as height, time, and velocity. It provides the framework for modeling the motion of projectiles with quadratic equations, where the vertical component is often expressed as a quadratic function of time.

Position Function

A position function describes the location of a moving object as a function of time. In the context of vertical projectile motion, this function is typically quadratic, reflecting the constant acceleration due to gravity. The position function allows one to compute where the projectile is at any given time and forms the basis for deriving other motion-related quantities.

Instantaneous Velocity

Instantaneous velocity refers to the rate at which an object's position is changing at a specific moment in time. It is mathematically defined as the derivative of the position function with respect to time. At the peak of a projectile's trajectory, the instantaneous velocity becomes zero because the upward motion transitions to downward motion, marking the moment of highest altitude.

Average Velocity

Average velocity is determined by dividing the change in position by the change in time over a given interval. It provides an approximation of the instantaneous velocity when the time interval is small. By computing the average velocity over successively smaller time intervals around a specific time, one can approximate the instantaneous velocity, which is the limit of these average velocities as the interval approaches zero.

Graphical Analysis

Graphical analysis involves interpreting the behavior of a function's graph to extract information about the motion of an object. In projectile motion, the graph of the position function is a parabola, and the vertex of the parabola indicates the maximum height. The point at which the graph reaches this maximum corresponds to the moment when the instantaneous velocity is zero. Additionally, by examining the slopes on different parts of the graph, one can determine intervals where the velocity is positive (ascending) or negative (descending).

Transcript

00:01 All right, so in this problem, we are given a function, which is s of t is equal to negative 16 t squared plus 128 plus 192.

00:11 Sorry, i'm a correction.

00:12 That's plus 128t.

00:14 All right, so it wants us to do, is it for part a? it wants us to make a graph.

00:20 And i have already done that for us.

00:23 You can just go ahead and plug that into your graph and calculator, dresmos, or whatever.

00:26 And you can see it's going to look something like this.

00:31 And it's not, i haven't put axes on it, just because the scale.

00:34 Is so big, but the graph you'll see it looks something like that.

00:38 He put it in desmos.

00:39 And then for part b, it wants us to find the time at which time equals a, at which the tangent line, the slope of the tangent line is equal to zero.

00:51 We can do this by first getting it into vertex form, standard vertex form.

00:57 And if we do that, well, the first step we're going to do is we're going to pull out a negative 16, which i've already done for us.

01:03 The second part, we're going to take half of that b term, 8.

01:06 So t minus 4 squared.

01:08 And then if we just had that in the middle, that wouldn't be equal to t squared minus a minus 12.

01:15 It would be equal to t minus 4 squared plus 16.

01:19 So to get that negative 12, we have to actually subtract the negative 28 there.

01:23 That's where the 28 comes from.

01:25 But that part is actually not that important because this already tells us all we need to know.

01:28 That t minus four is that position of the vertex.

01:32 And we know that the position, that wherever the vertex is, is wherever the slope of the tangent line is going to be zero or a quadratic equation.

01:40 So that t minus 4 tells us that at time equals 4, b protects, or time equals 4, the slope of the tangent line is equal to zero.

01:50 All right.

01:50 Then step c asks us to confirm this by making a table.

01:56 And i've already set up these intervals for us.

01:58 So the equation we're going to use to get that average velocity b bar is that it's going to be the value, the position at time 2 minus the position at 4.

02:06 Which we've already found is the position where we have zero slope.

02:11 And all over t2 minus 4.

02:13 So if we plug those values into our calculator, see over here we're going to start negative 16.

02:19 We're going to drop to negative 1 .6, then to negative 0 .16.

02:25 And finally, negative 0 .0 .6.

02:33 Yeah, okay, negative 0 .16...

Please give Ace some feedback