What is the approximate tangential speed of an object orbiting Earth with a radius of $1.8 \times 10^8 m$ and a period of 2.2 $\times 10^4$ s?
A. $7.7 \times 10^{-4} m / s$
B. $5.1 \times 10^4 m / s$
C. $7.7 \times 10^4 m / s$
D. $5.1 \times 10^5 m / s
Answer (2)
The tangential speed formula is v = T 2 π r .
Substitute r = 1.8 × 1 0 8 m and T = 2.2 × 1 0 4 s into the formula.
Calculate the tangential speed: v = 2.2 × 1 0 4 2 π ( 1.8 × 1 0 8 ) ≈ 51407.88 m / s .
The approximate tangential speed is 5.1 × 1 0 4 m / s .
Explanation
Problem Setup We are given the radius of an object's orbit around Earth, r = 1.8 × 1 0 8 m , and the period of its orbit, T = 2.2 × 1 0 4 s . We want to find the approximate tangential speed v of the object.
Formula for Tangential Speed The tangential speed of an object in circular motion is given by the formula: v = T 2 π r where r is the radius of the orbit and T is the period.
Substitution Now, we substitute the given values into the formula: v = 2.2 × 1 0 4 s 2 π ( 1.8 × 1 0 8 m ) v = 2.2 × 1 0 4 2 π × 1.8 × 1 0 8 m / s
Calculation Calculating the value: v ≈ 51407.88 m / s This is approximately 5.1 × 1 0 4 m / s .
Final Answer Comparing our calculated value to the given options, we see that the closest answer is 5.1 × 1 0 4 m / s .
Examples
Understanding tangential speed is crucial in many real-world scenarios. For example, when designing satellites orbiting Earth, engineers must calculate the precise speed required to maintain a stable orbit at a specific altitude. Similarly, in amusement park rides like spinning wheels, knowing the tangential speed helps ensure the safety and thrill of the riders. This concept also applies to understanding the motion of planets around the sun, where tangential speed varies depending on the planet's distance from the sun.
The approximate tangential speed of an object orbiting Earth with a radius of 1.8 × 1 0 8 m and a period of 2.2 × 1 0 4 s is calculated using the formula v = T 2 π r . This results in a tangential speed of approximately 5.1 × 1 0 4 m / s , making option B the correct choice. Therefore, the answer is B: 5.1 × 1 0 4 m / s .
;
