Answer:
the trajectory for the missile ![\mathbf{r(t) = \langle1000t,1400t, 368t -16t^2 \rangle}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%28t%29%20%3D%20%5Clangle1000t%2C1400t%2C%20368t%20-16t%5E2%20%5Crangle%7D)
the downrange of the missile R = 39570.6973 ft
Explanation:
Suppose the initial velocity v(0) = <1000, 1400, 368 > ft/sec
The single force which is acting over the missle is the gravitational force.
a (t) = <0, 0, -32> ft/sec²
We are to determine the trajectory and the downrange for the missle
Consider the motion of a given object r(t) to be:
![r(t) = v_it + \dfrac{1}{2}at^2](https://tex.z-dn.net/?f=r%28t%29%20%3D%20v_it%20%2B%20%5Cdfrac%7B1%7D%7B2%7Dat%5E2)
![r(t) = \langle1000,1400,368 \rangle t + \dfrac{1}{2} \langle 0,0, -32\rangle t^2](https://tex.z-dn.net/?f=r%28t%29%20%3D%20%5Clangle1000%2C1400%2C368%20%5Crangle%20t%20%2B%20%5Cdfrac%7B1%7D%7B2%7D%20%5Clangle%200%2C0%2C%20-32%5Crangle%20t%5E2)
![r(t) = \langle1000t,1400t,368t \rangle + \langle 0,0, -16 t^2 \rangle](https://tex.z-dn.net/?f=r%28t%29%20%3D%20%5Clangle1000t%2C1400t%2C368t%20%5Crangle%20%20%2B%20%5Clangle%200%2C0%2C%20-16%20t%5E2%20%5Crangle)
![r(t) = \langle1000t,1400t, 368t -16t^2 \rangle](https://tex.z-dn.net/?f=r%28t%29%20%3D%20%5Clangle1000t%2C1400t%2C%20368t%20-16t%5E2%20%5Crangle)
Thus, the trajectory for the missile ![\mathbf{r(t) = \langle1000t,1400t, 368t -16t^2 \rangle}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%28t%29%20%3D%20%5Clangle1000t%2C1400t%2C%20368t%20-16t%5E2%20%5Crangle%7D)
To determine the downrange of the missile,
![v(0) = \langle 1000,1400,368 \rangle](https://tex.z-dn.net/?f=v%280%29%20%3D%20%5Clangle%201000%2C1400%2C368%20%5Crangle)
where;
the horizontal vel. of the missle ![v_h= \sqrt{1000^2+1400^2}](https://tex.z-dn.net/?f=v_h%3D%20%5Csqrt%7B1000%5E2%2B1400%5E2%7D)
= 1720.4651 ft/s
the vertical vel. of the missile is ![v_v = 368 \ ft/s](https://tex.z-dn.net/?f=v_v%20%3D%20368%20%5C%20ft%2Fs)
The time required to reach the ground ![t =\dfrac{2 \times v_v}{g}](https://tex.z-dn.net/?f=t%20%3D%5Cdfrac%7B2%20%5Ctimes%20v_v%7D%7Bg%7D)
![t =\dfrac{2 \times 368}{32}](https://tex.z-dn.net/?f=t%20%3D%5Cdfrac%7B2%20%5Ctimes%20368%7D%7B32%7D)
t = 23 sec
Finally, the downrange of the missile ![R = v_h \times t](https://tex.z-dn.net/?f=R%20%3D%20v_h%20%5Ctimes%20t)
R = 1720.4651 × 23
R = 39570.6973 ft