Question

New York City is roughly $4800\mathrm{km}$ from Seattle. When it is 10:00 A.M. in Seattle, it is 1:00 P.M. in New York. Using this information, estimate (a) the rotational speed of the surface of Earth, (b) the circumference of Earth, and (c) the radius of Earth.

Short answer

(a) 1600 km/h, (b) 38400 km, (c) 6111 km.

Step-by-step solution

Convert the Time Difference into a Fraction of Earth's Rotation

The time difference between Seattle and New York is 3 hours. Since Earth completes a full rotation in 24 hours, the fraction of the earth's rotation corresponding to 3 hours is $\frac{3}{24}=\frac{1}{8}$. This fraction represents the angular separation between the two cities.

Calculate Earth's Rotational Speed

The distance between New York City and Seattle is given as $4800\text{ km}$. Since this distance corresponds to $\frac{1}{8}$ of Earth's circumference, the full circumference $C$ can be calculated by multiplying:$$C=4800\times 8=38400\text{ km}$$Earth makes a full rotation in 24 hours, so the rotational speed $v$ is:$$v=\frac{38400}{24}=1600\text{ km/h}$$.

Estimate the Circumference of Earth

From Step 2, we have already calculated the Earth's circumference as $38400\text{ km}$.

Calculate Earth's Radius Using the Circumference

The circumference of the Earth $C$ is related to the radius $r$ by the formula $C=2\pi r$. Rearranging for $r$, we get:$$r=\frac{C}{2\pi }=\frac{38400}{2\pi }\approx 6111\text{ km}$$.

Key concepts

Rotational Speed

The rotational speed of the Earth refers to how fast the Earth spins around its axis. To understand this concept, consider the scenario of traveling from Seattle to New York City. These two cities are aligned with different time zones, showing a time difference of 3 hours. This means, due to Earth's rotation, New York experiences sunlight 3 hours earlier than Seattle.
To calculate Earth's rotational speed, you start by determining the part of Earth's full rotation this time difference represents. Since there are 24 hours in a full day, a 3-hour difference is $\frac{1}{8}$ of a rotation. The actual distance between these cities is approximately 4800 km, indicating that this segment makes up $\frac{1}{8}$ of Earth's total circumference. Thus, the full circumference spans 38400 km.
Knowing the Earth spins completely in 24 hours helps us find the speed. By dividing the total circumference by the time, you can determine the rotational speed:

• Distance (full circumference): 38400 km
• Time for full rotation: 24 hours
• Rotational speed: $\frac{38400}{24}=1600$ km/h

This speed is crucial to understanding how Earth rotates on its axis, influencing not just time zones but also weather patterns and wind currents worldwide.

Earth's Circumference

The Earth's circumference is a key measurement that gives an overall sense of the planet's size. Circumference is the perimeter of a circle, and for Earth, it corresponds to the round shape encircling the Equator. From what's discussed about traveling between Seattle and New York City, a segment of distance between these cities represents $\frac{1}{8}$ of the total circumference of Earth.
If 4800 km makes up this fraction, to find the entire circumference, multiply:

• Segment length (1/8 of circumference): 4800 km
• Formula to find the total circumference: 4800 km $\times$ 8 = 38400 km

This comprehensive distance not only helps us understand large-scale geography but is essential in calculating other planetary characteristics.
With these measurements and applications, understanding Earth's circumference helps navigate the world and contributes to fields like global navigation, flight planning, and understanding great weather systems.

Earth's Radius

The radius of Earth allows us to appreciate the Earth's size from its center out to any point on the surface. The relationship between Earth's circumference and radius is used often in geometry. From the determined circumference of 38400 km, you can derive the radius using the formula for the circumference of a circle: $C=2\pi r$.
Rearranging the formula to solve for the radius $r$ is straightforward:

• Formula: $r=\frac{C}{2\pi }$
• Applying values: $r=\frac{38400}{2\pi }$ \approx\ 6111 km

The radius is a fundamental aspect when it comes to calculating Earth's volume and curvature. This calculation, while simple, reinforces the interconnectivity of Earth's measurements and helps scientists in studying Earth's physical properties.
Additionally, understanding Earth's radius aids in fields ranging from satellite deployment and telecommunications to understanding tectonic activities and geophysical modeling.