Flying into a headwind The velocity (in miles/hour) of an...

Flying into a headwind The velocity (in miles/hour) of an airplane flying into a headwind is given by $v(t)=30\left(16-t^{2}\right),$ for $0 \leq t \leq 3 .$ Assume that $s(0)=0.$ a. Determine and graph the position function, for $0 \leq t \leq 3.$ b. How far does the airplane travel in the first 2 hr? c. How far has the airplane traveled at the instant its velocity reaches $400 \mathrm{mi} / \mathrm{hr} ?$

Flying into a headwind The velocity (in miles/hour) of an airplane flying into a headwind is given by $v(t)=30\left(16-t^{2}\right),$ for $0 \leq t \leq 3 .$ Assume that $s(0)=0.$
a. Determine and graph the position function, for $0 \leq t \leq 3.$
b. How far does the airplane travel in the first 2 hr?
c. How far has the airplane traveled at the instant its velocity reaches $400 \mathrm{mi} / \mathrm{hr} ?$

Key Concepts

Algebraic Equation Solving in Kinematics

In problems where one must determine the time at which a given velocity is reached, algebraic techniques are used to solve the equation obtained by setting the velocity function equal to that specific velocity. This is critical in many kinematics problems to relate the time variable to other physical parameters.

Definite Integration for Accumulated Change

Definite integration is used to calculate the total change in a quantity over a given interval. In the context of motion, integrating the velocity function over a specified time period gives the total distance traveled by the object during that interval.

Graphing Functions to Visualize Behavior

Graphing functions such as the position function created from integrating velocity is a powerful tool for visualizing motion. It helps in understanding the trends, such as increases or decreases over time, and identifying important points like maximum or crossing points.

Initial Value Problems

When integrating to obtain a position function, an arbitrary constant appears which is determined by an initial condition (for instance, the starting position at time zero). This process forms an initial value problem that ensures the derived solution fits the specific situation described.

Integration as the Inverse Process of Differentiation

Integration is used to find the antiderivative of a given function; in kinematics, this allows us to determine the position function from the velocity function by integrating with respect to time. This process reveals how the accumulated effect of velocity yields the overall displacement, subject to an added constant of integration.

Transcript

00:01 Okay, so we're going to start this off.

00:04 So what we want to do first is we're going to be...

00:06 So we're looking at an equation.

00:09 This is an equation for the velocity of a plane moving in miles per hour.

00:13 It has a velocity that is given by the equation in terms of time, 30 times 16 minus t squared, and we know that at times zero, the distance or the position is zero as well.

00:24 So we're going to figure out what the position function is.

00:27 So we got s of t.

00:28 That's what we're going to live for.

00:30 To first figure that out.

00:31 Out.

00:31 What i'm going to do is i'm going to actually make this velocity equation into a, simplify it, or distribute this.

00:38 So i get 480 minus 30t squared.

00:43 Using that, i know that any time that i'm doing the antiderivative of the velocity, i get the position.

00:50 If i do that, all i'm doing is adding one to the exponent and dividing by that number.

01:01 So in this case, we divide by three.

01:03 In this case, we divide by one.

01:05 Don't remember.

01:06 Don't forget.

01:06 To add c.

01:08 Then what we're going to do is we're going to solve for c.

01:12 We know that at distance time equals zero, the distance is also equal to zero...

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