B A Doubt from Halliday Resnick Krane

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The discussion centers on the statement from Halliday Resnick Krane regarding the steeper descent of projectiles compared to their ascent due to drag force. It highlights that air resistance increases rapidly, leading to a sharp cut-off distance for thrown objects, such as paper wads or styrofoam peanuts, which fall almost vertically after reaching their peak. The behavior is explained through the equations of motion under linear drag, showing how velocity components change over time. The analysis emphasizes that gravity's effect becomes significant only after the initial speed dissipates due to drag. Understanding these dynamics is crucial for comprehending projectile motion in the presence of air resistance.
vibha_ganji
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Hello! In Chapter 4 of Halliday Resnick Krane on Pg. 73, they write “the descending motion is much steeper than the ascending motion” when discussing how the trajectory of a projectile changes when we consider drag force. Why is this statement true?
 
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You could think about terminal velocity in the vertical direction compared to terminal velocity in the horizontal direction.

Edit: Alternately, you could take a bunch of pieces of paper wad them up into a balls and throw them into the air at a 45 degree angle, both softly and as hard as you can. What I would expect you to observe is that there is a fairly sharp cut-off distance beyond which you cannot throw and that the paper wads will reach this distance and then fall more or less vertically to the floor. This is typical behavior for quadratic drag. Air resistance gets very large, very fast. Gravity only becomes significant once the high initial speed has had time to dissipate.

Styrofoam packing "peanuts" work more dramatically than paper wads to demonstrate this behavior.
 
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In the presence of linear drag, the equation of motion is
\begin{align*}
\dot{\mathbf{v}} + 2\gamma \mathbf{v} = \mathbf{g} \implies
\begin{cases}
\dot{v}_x + 2\gamma v_x &= 0 \\
\dot{v}_y + 2\gamma v_y &= -g
\end{cases}
\end{align*}A solution to the homogenous equation is ##e^{-2\gamma t}##. The forced equation for ##v_y## has a particular solution of ##-g/2\gamma##. The velocity is\begin{align*}
\mathbf{v}(t) = \begin{pmatrix} ae^{-2\gamma t} \\ be^{-2\gamma t} - \frac{g}{2\gamma} \end{pmatrix}
\end{align*}If ##\mathbf{v}(0) = (v_{\mathrm{0x}}, v_{\mathrm{0y}})## then ##a = v_{\mathrm{0x}}## and also ##b = g/2\gamma + v_{\mathrm{0y}}##, so that\begin{align*}
\mathbf{v}(t) &= e^{-2\gamma t} \begin{pmatrix} v_{\mathrm{0x}} \\ v_{\mathrm{0y}} + \frac{g}{2\gamma}[1- e^{2\gamma t}] \end{pmatrix}
\end{align*}Integrating, and putting ##\mathbf{x}(0) = 0##, gives\begin{align*}
\mathbf{x}(t) &= \frac{1}{2\gamma} (1- e^{-2\gamma t}) \begin{pmatrix} v_{\mathrm{0x}} \\ v_{\mathrm{0y}} + \frac{g}{2\gamma} \left[ 1 - \frac{2\gamma t}{1- e^{-2\gamma t}} \right] \end{pmatrix}
\end{align*}
 
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