/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 A jogger runs counterclockwise a... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 67

A jogger runs counterclockwise around a path of radius \(90.0 \mathrm{m}\) at constant speed. He makes 1.00 revolution in 188.4 s. At \(t=0,\) he is heading due east. (a) What is the jogger's instantaneous velocity at $t=376.8 \mathrm{s} ?\( (b) What is his instantaneous velocity at \)t=94.2 \mathrm{s} ?$

Short Answer

Expert verified
Answer: The instantaneous velocity of the jogger at \(t=376.8\,\text{s}\) is \(3.00\,\text{m/s}\) (due east) and at \(t=94.2\,\text{s}\) is \(3.00\,\text{m/s}\) (due west).

Step by step solution

01

Calculate the angular velocity of the jogger

Since the jogger completes 1.00 revolution in 188.4 seconds, we can find the angular velocity by using the formula: \(\omega = \frac{2\pi * n}{t}\), where \(n\) is the number of revolutions and \(t\) is the time taken. In this case, \(n = 1\) and \(t = 188.4\,\text{s}\): \(\omega = \frac{2\pi * 1}{188.4\,\text{s}}\) \(\omega = 0.0334\,\text{rad/s}\) The angular velocity of the jogger is \(0.0334\,\text{rad/s}\).
02

Find the linear velocity of the jogger

We can find the linear velocity (v) of the jogger by using the formula: \(v = r\omega\), where \(r\) is the radius of the path. Here, \(r = 90.0\,\text{m}\): \(v = 90.0\,\text{m} * 0.0334\,\text{rad/s}\) \(v = 3.00\,\text{m/s}\) The jogger's linear velocity is 3.00 m/s.
03

Determine the instantaneous velocity at \(t=376.8\,\text{s}\)

Since the jogger is moving counterclockwise around the path, we should note that at \(t=376.8\,\text{s}\), the jogger would have completed 2 revolutions and will be at the same position as at \(t=0\). Therefore, his instantaneous velocity will be due east: Instantaneous velocity at \(t=376.8\,\text{s}\): \(v = 3.00\,\text{m/s}\) (due east)
04

Determine the instantaneous velocity at \(t=94.2\,\text{s}\)

At \(t=94.2\,\text{s}\), the jogger would have completed half a revolution and will be headed due west. The magnitude of the velocity remains the same at \(3.00\,\text{m/s}\), but the direction changes: Instantaneous velocity at \(t=94.2\,\text{s}\): \(v = 3.00\,\text{m/s}\) (due west) In conclusion, the instantaneous velocity of the jogger at \(t=376.8\,\text{s}\) is \(3.00\,\text{m/s}\) (due east) and at \(t=94.2\,\text{s}\) is \(3.00\,\text{m/s}\) (due west).

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