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Projectile Motion: The x-value of an object moving under the principles of
projectile motion is x(theta, v_0, t) = (v_0*cos(theta))*t. A particular
projectile is fired with an intital velocity of v_0 = 250 ft/s and an angle of
elevation of theta = 60 degrees. It travels a distance of 375ft in 3 seconds.
Is the projectile more sensitive to errors in initial speed or angle of evelation?
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We are given the information below:

- \(v_0 = 250\)
- \(\theta = 60^{\circ}\)
- \(t = 3\)
- \(x(\theta, v_0, t) = (v_0cos(\theta))t\)

To see which variable is sensitive to errors, we compute the total differential for 3 variables defined by

\[ \begin{aligned} dz = x_\theta(\theta, v_0, t)d\theta + x_{v_0}(\theta, v_0, t)dv_0 + x_t(\theta, v_0, t)dt \end{aligned} \]

- Where the \(x_\theta, x_t\) and, \(x_{v_0}\) are the first order partial derivatives with respect to \(\theta\), t and \(v_0\) respectively. After computing the partial derivatives, we then plug in the numbers given to us and inspect which variables are subject to change.

\[ \begin{aligned} dz = x_\theta(\theta, v_0, t)d\theta + x_{v_0}(\theta, v_0, t)dv_0 + x_t(\theta, v_0, t)dt \\ = -v_0sin(\theta)t \,d\theta + cos(\theta)t \,d{v_0} + v_0cos(\theta) \,dt \\ =-750sin(60^\circ) \,d\theta + 3cos(60^\circ) \,d{v_0} + 250cos(60^\circ) \,dt \\ =-375\sqrt{3} \,d\theta + 1.5 \,d{v_0} + 125 \,dt \end{aligned} \]

To find whether the initial velocity \(v_0\) or angle of elevation \(\theta\) are the most sensitive, we notice that for small changes in \(d\theta\), dz is greatly reduced whereas if \(dv_0\) is changed in small increments, dz increases only slightly. Also note the coefficients of \(d\theta\) and \(dv_0\), \(d\theta\) coefficient is about 433 times larger than \(dv_0\) only in the case when the initial velocity is large.

Thus the projectile is more sensitive to errors in angle of elevation than the initial velocity.