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Chapter 2: Problem 18

In 3 h 24 min, a balloon drifts \(8.7 \mathrm{~km}\) north, \(9.7 \mathrm{~km}\) east, and \(2.9 \mathrm{~km}\) in elevation from its release point on the ground. Find (a) the magnitude of its average velocity and \((b)\) the angle its average velocity makes with the horizontal.

Short Answer

Expert verified
The average velocity of the balloon is both 0.0017 km/s in magnitude and directed at an angle of 10.71 degrees from the horizontal.

Step by step solution

01

Calculation of displacement vector

The balloon moves 8.7 km North, 9.7 km East, and 2.9 km Upward from its release point. Hence, the displacement vector \( \vec{d} \) is \{8.7 , 9.7 , 2.9\} km.
02

Conversion of time into proper unit

The time given is 3 hours 24 minutes. This needs to be converted into seconds in order to have a consistent set of units. \(3 h = 3 \times 3600 s = 10800 s\), \(24 min = 24 \times 60 s = 1440 s\), hence, total time \(t = 10800 s + 1440 s = 12240 s\).
03

Calculation of magnitude of average velocity

To find the magnitude of the average velocity we use the formula for average velocity \(v_{avg} \), which is the displacement over time. Therefore, magnitude of average velocity \(v_{avg}\) is \( \sqrt{(8.7^2 + 9.7^2 + 2.9^2)} km / 12240 s = 0.0017 km/s \).
04

Calculation of direction of average velocity

To find the direction of the average velocity, we make use of the inverse tangent function (also known as arctan) applied to the height over the horizontal displacement. Let the direction be \(θ\). \(θ = atan(2.9 / \sqrt{(8.7^2 + 9.7^2)}) = 10.71 degrees \).

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Most popular questions from this chapter

The position of a particle moving in an \(x y\) plane is given by \(\overrightarrow{\mathbf{r}}=\left[\left(2 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3}-(5 \mathrm{~m} / \mathrm{s}) t\right] \hat{\mathbf{i}}+\left[(6 \mathrm{~m})-\left(7 \mathrm{~m} / \mathrm{s}^{4}\right) t^{4}\right] \hat{\mathbf{j}} . \quad\) Cal- culate \((a) \overrightarrow{\mathbf{r}},(b) \overrightarrow{\mathbf{v}}\), and \((c) \overrightarrow{\mathbf{a}}\) when \(t=2 \mathrm{~s}\).

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