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Chapter 3: Problem 8

\({ }^{*}\) By expanding the logarithm in (3.17), find the approximate equation for the trajectory of a projectile subject to small atmospheric drag to first order in \(\gamma\). (Note that this requires terms up to order \(\gamma^{3}\) in the logarithm.) Show that to this order the range (on level ground) is $$ x=\frac{2 u w}{g}-\frac{8 \gamma u w^{2}}{3 g^{2}} $$ and hence that to maximize the range for given launch speed \(v\) the angle of launch should be chosen to satisfy \(\cos 2 \alpha=\sqrt{2} \gamma v / 3 g .\) (Hint: In the term containing \(\gamma\), you may use the zeroth-order approximation for the angle.) For a projectile whose terminal speed if dropped from rest (see Chapter 2, Problem 13) would be \(500 \mathrm{~m} \mathrm{~s}^{-1}\), estimate the optimal angle cand the range if the launch speed is \(100 \mathrm{~m} \mathrm{~s}^{-1}\).

Short Answer

Expert verified
The final goal is to find the optimal launch angle and range for a projectile with given input parameters (launch speed and terminal speed constraint), considering a small atmospheric drag. We will use the formula \(\cos 2 \alpha = \frac{\sqrt{2} \gamma v}{3g}\) to verify our result.

Step by step solution

01

Expand the logarithm in equation (3.17)

Using the well-known logarithmic expansion, which is given by: $$ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots $$ We can expand equation (3.17) up to order \(\gamma^{3}\). Note that we are only interested in the first-order terms in \(\gamma\).
02

Find the approximate equation for trajectory

After expanding the logarithm, we can find the approximate equation for the trajectory of the projectile. Keep the terms up to the order of \(\gamma\) in the equation and ignore higher-order terms.
03

Find the formula for range

Now, we can use the trajectory equation to find the formula for the range (\(x\)) on level ground. Remember that we want to find an expression containing both \(u\) (the horizontal component of the initial velocity) and \(w\) (the vertical component of the initial velocity).
04

Find the optimal launch angle

To find the optimal launch angle, we need to maximize the range (\(x\)) with respect to the launch angle (\(\alpha\)). To do this, we should differentiate \(x\) with respect to \(\alpha\) and set it equal to 0. After finding the optimal angle, use the formula $$ \cos 2 \alpha = \frac{\sqrt{2} \gamma v}{3g} $$ to verify our result.
05

Find the optimal angle and range for given input parameters

Finally, we can find the optimal launch angle and range for the given input parameters. The launch speed is \(100~m~s^{-1}\), and the terminal speed constraint is \(500~m~s^{-1}\). To find the range (\(x\)), we will use the formula obtained in Step 3, keeping in mind the optimal launch angle obtained in Step 4.

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