91Ó°ÊÓ

Cheetahs running at top speed have been reported at an astounding \(114 \mathrm{~km} / \mathrm{h}\) (about \(71 \mathrm{mi} / \mathrm{h})\) by observers driving alongside the animals. Imagine trying to measure a cheetah's speed by keeping your vehicle abreast of the animal while also glancing at your speedometer, which is registering \(114 \mathrm{~km} / \mathrm{h}\). You keep the vehicle a constant \(8.0 \mathrm{~m}\) from the cheetah, but the noise of the vehicle causes the cheetah to continuously veer away from you along a circular path of radius \(92 \mathrm{~m}\). Thus, you travel along a circular path of radius \(100 \mathrm{~m} .\) (a) What is the angular speed of you and the cheetah around the circular paths? (b) What is the linear speed of the cheetah along its path? (If you did not account for the circular motion, you would conclude erroneously that the cheetah's speed is \(114 \mathrm{~km} / \mathrm{h}\), and that type of error was apparently made in the published reports)

Step by step solution

01

Convert Cheetah Speed

Begin by converting the reported linear speed of the cheetah from kilometers per hour to meters per second for use in subsequent calculations. Use the conversion factor: \[\text{Speed} = 114 \frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 31.67 \frac{\text{m}}{\text{s}}\] So, the linear speed of the cheetah, considering a straight line, is 31.67 m/s.

02

Determine Angular Speed

The angular speed (\(\omega\)) is calculated as the linear speed (\(v\)) divided by the radius (\(r\)) of the circular path. For the observer, \[\omega = \frac{v}{r} = \frac{31.67 \text{ m/s}}{100 \text{ m}} = 0.3167 \text{ rad/s}\] Therefore, the angular speed of the observer and the cheetah around the circular paths is 0.3167 rad/s.

03

Calculate Linear Speed of Cheetah

To find the actual linear speed along its curved path, use the angular speed calculated previously and multiply it by the radius of the cheetah's path. \[v_{\text{cheetah}} = \omega \times r_{\text{cheetah}} = 0.3167 \text{ rad/s} \times 92 \text{ m} = 29.14 \text{ m/s}\] Thus, the linear speed of the cheetah along its circular path is 29.14 m/s.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Speed

Angular speed is a measure of how fast an object moves through an angle, generally showing how fast it travels around a circle. In essence, it tells us how quickly something spins or rotates.
Angular speed is typically expressed in radians per second, a unit that signifies angles and is derived from the motion along a circular path.
For this scenario, imagine the cheetah and the observer moving in circles. The angular speed would then be calculated by dividing the linear speed by the radius of the respective circular path.

• To find the observer's angular speed, you divide their linear speed (31.67 m/s) by their orbit radius (100 m).
• The result is 0.3167 radians per second, indicating how quickly they complete a full circle with reference to time.

This value provides a better understanding of the cheetah's motion than just using the straight-line speed of 114 km/h obtained from the speedometer in a non-circular perspective.

Linear Speed

Linear speed refers to how fast an object moves along a path in a straight line. When considering motion in a circular path, as in this scenario, it's crucial to differentiate between linear speed along the curve and the straightforward speed shown by a vehicle's speedometer.
Linear speed, in simpler terms, is just the regular speed we know and measure daily, like when driving a car.
For the cheetah, however, the linear speed along its circular path is different and requires adjusting the often mistaken value of 114 km/h shown on a straight path.

• When initially converted, the linear speed straight-line was 31.67 m/s.
• However, when moving along its path, considering the radius of 92 m, the calculated linear speed is 29.14 m/s, showing a correction made for the path's curvature.

This helps correct errors in measurements or reports about speeds from earlier settings aimed solely at straight paths.

Circular Motion

Circular motion describes how objects move in curves, clearly distinguished from straight-line movement.
Understanding circular motion involves comprehending how forces, such as centripetal force, keep objects in such paths.
Often, circular motion is described through angular and linear speeds, as seen with the cheetah and the observer in this context.

• The cheetah moves with a radius of 92 meters while the observer follows a 100-meter radius path.
• Both entities' speeds are adjusted for their curved paths, significantly resigning on angular and linear dynamics.

This special consideration of circular motion helps identify true object speeds and paths, important for accurate motion comprehension beyond simplified linear approximations.