91Ó°ÊÓ

The Pythagorean theorem is a fundamental principle in geometry, particularly when dealing with right triangles. It states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. This can be expressed in the formula: \[ c^2 = a^2 + b^2 \]where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.
In our exercise with the meteoroid, the Pythagorean theorem helps us calculate its overall speed by treating the eastward and downward motions as perpendicular components.
This means that the meteoroid's total speed is like the hypotenuse of a right triangle formed by these two velocity components.

• The eastward movement is the base of this right triangle.
• The downward motion is the height of the triangle.

Using this theorem allows us to combine these perpendicular velocities into a single speed value.

In physics, particularly in kinematics, motion components refer to breaking complex motion into simpler parts, usually along perpendicular axes. In the case of the meteoroid, its motion is divided into two components: one horizontal and one vertical.
The horizontal component for the meteoroid moving east is given as \(18.3 \, \text{km/s}\). This represents the speed along the eastward direction. Similarly, the vertical component, which is its speed while descending, is \(11.5 \, \text{km/s}\). By treating these components separately, we gain a clear understanding of how the meteoroid is traveling through space.
This division is incredibly useful because it allows us to use mathematical tools like the Pythagorean theorem to combine them back into a single value when needed, such as finding the actual velocity.

Calculating speed involves determining how fast an object is moving irrespective of its direction. For our meteoroid problem, speed calculation involves combining the horizontal and vertical motion components.
The formula derived from the Pythagorean theorem gives us the meteoroid's speed: \[ v = \sqrt{v_x^2 + v_y^2} \]where \(v_x\) is the horizontal component (eastward) and \(v_y\) is the vertical component (downward).
Inserting the known values:

• Eastward speed: \(v_x = 18.3 \, \text{km/s}\)
• Downward speed: \(v_y = 11.5 \, \text{km/s}\)

We calculate:

• \(v_x^2 = 334.89\)
• \(v_y^2 = 132.25\)

Adding these gives \(467.14\). Finally, taking the square root results in the meteoroid's speed: \(21.62 \, \text{km/s}\).

A right triangle is a type of triangle where one of the angles is exactly 90 degrees. The relationship between its sides is described by the Pythagorean theorem, making it a crucial tool for many scientific calculations, including motion-related problems.
In the setting of our meteoroid problem, visualizing the horizontal and vertical components as the two shorter sides of a right triangle helps you understand how these independent motions combine into a single trajectory. The hypotenuse then represents the meteoroid's total speed, which we solve for using the Pythagorean theorem.
Understanding the concept of a right triangle not only gives us a great visual aid but also provides the necessary mathematical grounding for effectively combining the motion components.