The velocity (in mil / hr) of an airplane flying into a...

The velocity (in mil / hr) 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 $s(0)=0$ and $t$ is measured in hours. a. Determine 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 $400 \mathrm{mi} / \mathrm{hr} ?$

The velocity (in mil / hr) 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 $s(0)=0$ and $t$ is measured in hours.
a. Determine 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

Initial Value Problem

When obtaining the position function through integration, a constant of integration appears. The initial condition (such as the initial position) is then used to determine this constant, ensuring that the resulting function accurately represents the position over time.

Definite Integration

Definite integrals are employed to calculate the total distance traveled over a specific time interval. By integrating the velocity function between two time points, one can find the net displacement, which in cases where velocity is always positive, corresponds to the total distance.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, allowing one to recover a position function by integrating the velocity function. It is central in understanding how an accumulation process (like distance traveled) can be represented as an antiderivative of a rate of change (like velocity).

Integration of a Velocity Function

Integration is used to find the position function from a given velocity function by computing the antiderivative. This process converts a rate function (velocity) into an accumulation function (position), effectively capturing the total displacement over time.

Solving for Specific Time Instances

This involves setting the velocity function equal to a given value and solving for the time variable. This step is crucial for identifying when certain events occur, such as when the airplane reaches a predetermined speed.

Transcript

00:01 We have a plane flying into a headwind.

00:03 We are given its velocity, and we're told its position at the time of zero is zero.

00:09 First, we want to figure out its position function.

00:14 So the position function is the integral of the velocity function.

00:23 So, and i'll go ahead and distribute the 30 in that velocity function.

00:27 So i have 480 minus 30 t square.

00:32 The integral of that would be 480 t minus 10 t cubed plus a constant.

00:48 C we can find because we know the position at zero seconds is zero.

00:54 So if i fill in zero for s of t and zero for the time, i can see that c is zero.

01:05 So if i remove the c value, the position function we have is 480 t minus 10 t cubed.

01:23 Now let's look at part b.

01:27 Second part of our question wants to know how far the airplane travels in two hours.

01:35 So we're going to take the position at two hours and subtract from it its position at zero hours.

01:45 Now, normally we'd have to worry about if there was a direction change, but if we look at the original formula, that's going to be a positive as long as t is at least four, and we just want this for the first two hours.

02:01 As of two, if i fill that into my position equation, i get 960 minus 80.

02:11 Subtract s of zero, but that's just zero...

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