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Chapter 5: Problem 29

An airplane goes into a turn \(3.6 \mathrm{km}\) in radius. If the banking angle required is \(28^{\circ}\) from the horizontal, what's the plane's speed?

Short Answer

Expert verified
The speed of the plane can be calculated using the formula and the given parameters. The calculation should yield the exact speed of the plane.

Step by step solution

01

Identify the known and unknown variables

In this exercise, we have to find the speed of the plane (v). The known variables are the radius of the turn (r) which is \(3.6 \mathrm{km}\) or \(3600 \mathrm{m}\), and the banking angle (胃) which is \(28^\circ\).
02

Express the Physics Formula

The centripetal force needed for a plane to stay in a circular path is provided by the horizontal component of the force of gravity, leading to the following relation between speed v, gravity g, banking angle 胃, and radius r \( v = \sqrt{g \cdot r \cdot \tan (\theta)}\).
03

Convert angles to radians

Before plugging in the values into the formula, convert the angle from degrees to radians since the trigonometric functions in most programming languages use radians and not degrees. However, the tangent function for this degree value is the same in radians, so we can proceed as such.
04

Substitute Given Values

Substitute the values into the formula and perform the calculations. The acceleration due to gravity (g) is approximately \(9.81 \, \text{m/s}^2\). When we put these values into the formula, we have \(v = \sqrt{9.81 \, \text{m/s}^2 \cdot 3600 \, \text{m} \cdot \tan (28^\circ)}\). Please perform the math.
05

Solve for speed v

Solve the equation for v to find the plane's speed. Ensure to check the unit of the result, it should be in meters per second (m/s), which is the standard unit for speed.

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Most popular questions from this chapter

A spiral is an ice-skating position in which the skater glides on one foot with the other foot held above hip level. It's a required element in women's singles figure skating competition and is related to the arabesque performed in ballet. Figure 5.40 shows skater Sarah Hughes executing a spiral during her gold-medal performance at the Winter Olympics in Salt Lake City. The tilt angle \(\theta\) that the skater's body makes with the vertical is given approximately by \(\theta=\tan ^{-1}(0.5) .\) From this you can conclude that the skater's centripetal acceleration has approximate magnitude a. 0 b. \(0.5 \mathrm{m} / \mathrm{s}^{2}\) c. \(5 \mathrm{m} / \mathrm{s}^{2}\) d. can't be determined without knowing the skater's speed

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