During takeoff, an airplane's angle of ascent is 18^∘ and its speed is 260 feet per second. (a)

During takeoff, an airplane's angle of ascent is 18^∘ and...

During takeoff, an airplane's angle of ascent is $18^{\circ}$ and its speed is 260 feet per second.
(a) Find the plane's altitude after 1 minute.
(b) How long will it take for the plane to climb to an altitude of $10,000$ feet?

Key Concepts

Vector Decomposition

Vector decomposition involves breaking a vector, like an airplane's velocity, into its horizontal and vertical components. This is essential when analyzing motion at an angle because it lets us isolate the vertical component which directly affects altitude, while the horizontal component relates to distance covered.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are used to relate angles to ratios of sides in right-angled triangles. They allow us to compute the components of vectors, making it possible to determine how much of a given velocity is directed vertically or horizontally.

Unit Conversion

Unit conversion is the process of converting measurements from one system to another to ensure consistency in calculations. In problems involving motion, converting time units (like minutes to seconds) is often necessary to align the units of velocity and time.

Linear Kinematics

Linear kinematics deals with the motion of objects in a straight line, often under constant velocity or acceleration. In the context of calculating altitude, the vertical component of the velocity acts as a constant rate of climb, making it a kinematics problem where displacement is found by multiplying this rate by time.

Transcript

00:01 So we're told that a plane is going 260 feet per second at an angle of ascent of 18 degrees, and we're going to try and find the altitude of this plane after one minute or 60 seconds.

00:13 So the length of this hypotenuse is going to be the distance traveled by the plane, which in this case will be 260 feet per second times however many seconds we've gone.

00:24 So in this case it will be 60.

00:25 So if we calculate that, we get 260 feet per second.

00:30 Second multiplied by 60 seconds.

00:35 The seconds will cancel out and we're left with feet.

00:39 And our answer is 15 ,600 for the hypotenuse of our triangle.

00:46 And now we can just use a trigonomic function to actually find this x value.

00:50 And we're going to want to use one that involves the opposite as well as the hypotenuse side.

00:55 And in that case, we'll be using sign since so catoa, sign is opposite over hypotenuse.

01:01 So you can say the sign of 18 degrees is equal to the opposite side length is the length we're trying to find.

01:11 So x divided by 15 ,600.

01:14 And now we can multiply by 15 ,600 on both sides.

01:18 So we get 15 ,600.

01:20 Sign of 18 degrees is equal to x.

01:25 And if you plug this into a calculator, this is equal to 4 ,820 .7 feet.

01:35 So the altitude is 4 ,820 .7 feet of our airplane after one minute.

01:44 And in part b, what we're going to try and find is how long it takes for our plane to actually reach 10 ,000 feet...

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