Answer:
a) The bullet hits the ground after 64 seconds.
b) The bullet hits the ground 113,511.7 feet away.
c) The maximum height attained by the bullet is of 16,384 feet.
Step-by-step explanation:
Equations of motion:
The equations of motion for the bullet are:
![x(t) = (v_0\cos{\alpha})t](https://tex.z-dn.net/?f=x%28t%29%20%3D%20%28v_0%5Ccos%7B%5Calpha%7D%29t)
![y(t) = (v_0\sin{\alpha})t - 16t^2](https://tex.z-dn.net/?f=y%28t%29%20%3D%20%28v_0%5Csin%7B%5Calpha%7D%29t%20-%2016t%5E2)
In which
is the initial speed and
is the angle.
Initial speed of 2048 ft/s at an angle of 30o to the horizontal.
This means that
.
So
![x(t) = (v_0\cos{\alpha})t = (2048\cos{30})t = 1773.62t](https://tex.z-dn.net/?f=x%28t%29%20%3D%20%28v_0%5Ccos%7B%5Calpha%7D%29t%20%3D%20%282048%5Ccos%7B30%7D%29t%20%3D%201773.62t)
![y(t) = (v_0\sin{\alpha})t - 16t^2 = (2048\sin{30})t - 16t^2 = 1024t - 16t^2](https://tex.z-dn.net/?f=y%28t%29%20%3D%20%28v_0%5Csin%7B%5Calpha%7D%29t%20-%2016t%5E2%20%3D%20%282048%5Csin%7B30%7D%29t%20-%2016t%5E2%20%3D%201024t%20-%2016t%5E2)
(a) After how many seconds will the bullet hit the ground?
It hits the ground when
. So
![1024t - 16t^2 = 0](https://tex.z-dn.net/?f=1024t%20-%2016t%5E2%20%3D%200)
![16t^2 - 1024t = 0](https://tex.z-dn.net/?f=16t%5E2%20-%201024t%20%3D%200)
![16t(t - 64) = 0](https://tex.z-dn.net/?f=16t%28t%20-%2064%29%20%3D%200)
16t = 0 -> t = 0 or t - 64 = 0 -> t = 64
The bullet hits the ground after 64 seconds.
(b) How far from the gun will the bullet hit the ground?
This is the horizontal distance, that is, the x value, x(64).
![x(64) = 1773.62(64) = 113511.7](https://tex.z-dn.net/?f=x%2864%29%20%3D%201773.62%2864%29%20%3D%20113511.7)
The bullet hits the ground 113,511.7 feet away.
(c) What is the maximum height attained by the bullet?
This is the value of y when it's derivative is 0.
We have that:
![y^{\prime}(t) = 1024 - 32t](https://tex.z-dn.net/?f=y%5E%7B%5Cprime%7D%28t%29%20%3D%201024%20-%2032t)
![1024 - 32t = 0](https://tex.z-dn.net/?f=1024%20-%2032t%20%3D%200)
![32t = 1024](https://tex.z-dn.net/?f=32t%20%3D%201024)
![t = \frac{1024}{32} = 32](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B1024%7D%7B32%7D%20%3D%2032)
At this instant, the height is:
![y(32) = 1024(32) - 16(32)^2 = 16384](https://tex.z-dn.net/?f=y%2832%29%20%3D%201024%2832%29%20-%2016%2832%29%5E2%20%3D%2016384)
The maximum height attained by the bullet is of 16,384 feet.