A gun at O fires shells with an initial speed of 200m/s but with a variable angle of inclination α. Take g=10 m/s^2.
(i) If a fortress F is situated 2km away on top of a mountain 1000 metres high show that if the gun is set at an angle of 45 degrees the fortress will be hit.
(ii) Find the flight time and impact of speed of the shell.
I'm after some help with this projectile motion problem. I just need to know where to slot things into a formula and differentiation.
Apply the equations below,
$$x = v_xt,\>\>\>\>\> y = v_y t -\frac12 g t^2$$
Substitute the givens to get,
$$2000 = 200\cdot \cos 45 \cdot t, \>\>\>\>\>\>y = 200\cdot\sin 45\cdot t - \frac12\cdot 10\cdot t^2$$
Solve for $y$ and $t$,
$$y=1000,\>\>\>\>\> t = 10\sqrt2$$
which shows the gun hits the target at the height of 1000 meters and the time of flight is $10\sqrt2$ seconds.
The vertical speed at the impact is $v_y-gt = 200\cdot \sin 45 - 10\cdot 10\sqrt2=0$. So, the impact speed of the shell is $v_x= 200\cdot \cos 45 = 100\sqrt2$m/s.
