In a proof-of-concept experiment for an antiballistic missile defense system, a missile is fired from the ground of a shooting range toward a stationary target on the ground. The system detects the missile by radar, analyzes in real time its parabolic motion, and determines that it was fired from a distance \(x_{0}=5000 \mathrm{~m}\), with an initial speed of \(600 \mathrm{~m} / \mathrm{s}\) at a launch angle \(\theta_{0}=20^{\circ} .\) The defense system then calculates the required time delay measured from the launch of the missile and fires a small rocket situated at \(y_{0}=500 \mathrm{~m}\) with an initial velocity of \(v_{0} \mathrm{~m} / \mathrm{s}\) at a launch angle \(\alpha_{0}=60^{\circ}\) in the \(y z\) -plane, to intercept the missile. Determine the initial speed \(v_{0}\) of the intercept rocket and the required time delay.

Step by step solution

01

Convert angles to radians

We'll need to work with angles in radians. Convert the given angles from degrees to radians using the following conversion factor: \(1 \, radian = \frac{180}{\pi} \, degrees\). $$ \theta_{0} = 20^{\circ} = \frac{20 \pi}{180} = \frac{\pi}{9} \, radians $$ $$ \alpha_{0} = 60^{\circ} = \frac{60 \pi}{180} = \frac{\pi}{3} \, radians $$

02

Write the equations for the horizontal and vertical positions of the two objects

For the missile (denoted by subscript \(\textit{m}\)) and the intercept rocket (denoted by subscript \(\textit{r}\)), write the equations for their horizontal (\(x\)) and vertical (\(y\)) positions as functions of time: Missile: $$ x_{m}(t) = x_{0} + v_{m0} \cos \theta_{0} t $$ $$ y_{m}(t) = v_{m0} \sin \theta_{0} t - \frac{1}{2} g t^2 $$ Intercept Rocket: $$ x_{r}(t + \Delta{t}) = v_{r0} \cos \alpha_{0} (t + \Delta{t}) $$ $$ y_{r}(t + \Delta{t}) = y_{0} + v_{r0} \sin \alpha_{0}(t + \Delta{t}) - \frac{1}{2} g (t + \Delta{t})^2 $$ Where \(g\) is the acceleration due to gravity, \(\Delta{t}\) is the time delay, and \(v_{m0}\) and \(v_{r0}\) are their respective initial velocities.

03

Set the horizontal positions equal

Since the two objects need to intersect horizontally, set \(x_{m}(t)\) equal to \(x_{r}(t + \Delta{t})\) and solve for \(t\). Use the given values: $$ 5000+600 \cos \frac{\pi}{9} t= v_{r0}\cos\frac{\pi}{3}(t+\Delta{t}) $$

04

Set the vertical positions equal

Similarly, set \(y_{m}(t)\) equal to \(y_{r}(t + \Delta{t})\): $$ 600 \sin \frac{\pi}{9} t - \frac{1}{2} g t^2 = 500 + v_{r0} \sin \frac{\pi}{3} (t + \Delta{t}) - \frac{1}{2} g (t + \Delta{t})^2 $$

05

Solve for time delay and initial speed of the intercept rocket

Now we have a system of equations with the two unknowns \(v_{r0}\) and \(\Delta{t}\). Solve for these two unknowns using any method (like substitution or computer algebra system). Note: Since the resulting equations are cumbersome, it's advised to use a computer algebra system or a numerical simulation tool to solve the system of equations. The result will give the initial speed \(v_{0}\) of the intercept rocket and the required time delay \(\Delta{t}\).

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