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?

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

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