A missile is launched in your direction. Using a range finder, you measure its altitude at the t

A missile is launched in your direction. Using a range...

Key Concepts

Kinematics

Kinematics is a branch of mechanics dedicated to describing motion without considering the forces causing it. It provides the mathematical framework concerning displacement, velocity, acceleration, and time, which are integral for modeling trajectories and predicting events in motion-based systems.

Projectile Motion

Projectile motion describes the movement of an object thrown or propelled into the air, influenced only by gravity (and possibly air resistance). Its path is typically modeled by quadratic equations, which capture key aspects such as initial velocity, acceleration due to gravity, and total flight duration.

Curve Fitting

Curve fitting involves selecting a mathematical model—often a polynomial or another functional form—that best approximates a set of observed data points. This technique is essential for understanding and predicting the behavior of a system from discrete measurements, such as determining a projectile's path based on altitude readings over time.

Extrapolation

Extrapolation is the process of estimating values beyond the range of the given data based on established trends in the data. This concept is crucial in scenarios where predicting future events or outcomes is necessary, like determining the moment when a moving object reaches a certain positional threshold.

Transcript

00:01 This question reads, a missile's trajectory takes it to a maximum altitude of 12 ,000 kilometers.

00:16 So the altitude is interesting.

00:21 Notice that that's a maximum altitude.

00:25 12, not 12 ,000, 1 ,200 kilometers.

00:33 So that would be 1 ,200 ,000 meters.

00:38 Notice that it says that it's maximum altitude.

00:40 That means that its velocity when it reaches that altitude.

00:45 So i'm going to write v.

00:47 Sub final is zero.

00:51 It's launch speeds.

00:53 That's v initial.

00:56 Is 6 ,100 meters per second.

01:04 And again, it goes to 1 ,200 kilometers, which is 1 ,200 ,000 meters.

01:10 How fast is it moving at the peak of its trajectory? okay, so perhaps that's telling us that the peak of its trajectory is not zero.

01:32 So we'll put a question mark there.

01:35 Okay.

01:37 So we can write kinetic energy initial plus potential energy initial equals kinetic energy final, plus potential energy final.

01:54 So one half m v initial.

02:01 Okay, we do have the initial.

02:03 So i'm going to write v.

02:09 Minus g m little m over our initial well the initial radius is the radius of the earth the initial distance is the radius of the earth because it starts from earth equals k final one half m v squared so that's what we're trying to figure out i'm going to put a sub f there even though it's a little bit hard to write on this whiteboard um and then minus g m, little m, over r final.

02:49 Well, our final, we're given an altitude, so this would have to be the radius of the earth plus the altitude.

02:58 Okay, so we just need to solve for v final.

03:03 So first thing that i'm going to do is i'm going to add this term to both sides.

03:08 So that's going to give me one -half m v .i squared plus.

03:18 I'm also going to factor out the gm -m -1 over r -sub -e plus h minus 1 over r -sub -e.

03:41 And that equals 1 -half m v.

03:49 Squared...

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