Question

In gym class you run $22\mathrm{m}$ horizontally, then climb a rope vertically for $4.8\mathrm{m}$. What is the direction angle of your total displacement, as measured from the horizontal?

Short answer

The direction angle is approximately 12.34 degrees from the horizontal.

Step-by-step solution

Identify Components

First, recognize your movement vector components. You run 22 meters horizontally and climb 4.8 meters vertically. Hence, you have two vectors: horizontal ( 22 m) and vertical ( 4.8 m).

Use Trigonometry to Find Direction

To find the direction angle $\theta$, you use the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle: $\tan \theta =\frac{\text{vertical component}}{\text{horizontal component}}=\frac{4.8}{22}$.

Calculate the Angle

Now, solve for $\theta$ by taking the arctangent (inverse tangent) of the ratio: $\theta =\arctan \left(\frac{4.8}{22}\right)$. Calculate it using a calculator to get the angle in degrees.

Key concepts

Trigonometry

Trigonometry is a field of mathematics that deals with the relationships between the sides and angles of triangles. It is essential in solving problems related to right triangles, like the exercise we are looking at.
Trigonometric functions such as sine, cosine, and tangent are used to relate angles to the ratios of a triangle's sides. For our specific example, the tangent function is most relevant.

• The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. For angle $\theta$, the tangent is expressed as $\tan (\theta )=\frac{\text{opposite side}}{\text{adjacent side}}$.
• These trigonometric principles allow us to find unknown angles or sides given certain known dimensions.

By setting up equations based on these functions, you can solve for unknowns like the direction angle in our exercise.

Vector Components

Vectors are quantities having both magnitude and direction. In our scenario, the movement comprises two main vector components: a horizontal run and a vertical climb.

• The horizontal component measures the distance traveled along the horizontal plane, which is $22\text{ m}$ in this exercise.
• The vertical component is the distance traveled vertically, amounting to $4.8\text{ m}$.

Understanding these components helps in calculating resultant vectors and their direction angles. When displacement involves more than one vector, breaking them into components simplifies the analysis. We often use trigonometric functions on these components to understand the overall displacement angle, as demonstrated here.

Right Triangle

A right triangle is a type of triangle that has one angle measuring ${90}^{\circ }$. It is fundamental in connecting the principles of trigonometry and vector analysis.
In this exercise, the horizontal and vertical components form the two perpendicular sides of a right triangle.

• The hypotenuse is the resultant displacement, which can be analyzed to deduce overall direction and magnitude.
• The right angle aids in employing trigonometric functions effectively, such as using the tangent function to determine angle $\theta$.

Utilizing the properties of right triangles allows us to calculate unknowns with precision, providing a mathematical framework that is both powerful and widely applicable in physics and engineering problems.