i. Well, the greatest elevation above the ground reached by the projectile is the highest point it reaches during its flight. And if I had a dollar for every time I reached new heights, I'd have at least three dollars! Anyway, to find this point we can use the projectile motion equations. We know the initial velocity (180 m/s) and the launch angle (30 degrees), so we can split the velocity into horizontal and vertical components. The vertical component of velocity is 180 m/s * sin(30), which is 90 m/s. Now, we can use the kinematic equation for vertical motion:
v^2 = u^2 + 2as
Where v is the final vertical velocity (0 m/s at the top), u is the initial vertical velocity (90 m/s), a is the acceleration due to gravity (-9.8 m/s^2), and s is the displacement (which we want to find).
Substituting the known values, we get:
0^2 = (90)^2 + 2*(-9.8)*s
Simplifying further, we find:
-9.8s = -4050
Solving for s, we find that the displacement is approximately 414.3 meters.
So, the greatest elevation above the ground reached by the projectile is 414.3 meters!
ii. Now let's find the horizontal distance from the gun to the point where the projectile strikes the ground. This can be found using the horizontal component of the initial velocity, which is 180 m/s * cos(30), which is approximately 155.9 m/s.
Now, we can use the kinematic equation for horizontal motion:
s = ut
Where s is the horizontal displacement, u is the initial horizontal velocity, and t is the time of flight.
To find the time of flight, we can use the equation:
s = ut + (1/2)at^2
Since the horizontal acceleration is 0 (no horizontal forces acting on the projectile), the equation simplifies to:
s = ut
Substituting the known values, we have:
s = (155.9)(t)
To find t, we can use the vertical motion equation:
s = ut + (1/2)at^2
But this time, we're looking for the time it takes for the projectile to hit the ground, so the final vertical displacement is 0.
Thus, we have:
0 = (90)t + (1/2)(-9.8)t^2
Simplifying further, we find:
-4.9t^2 + 90t = 0
Factoring out t, we get:
t(-4.9t + 90) = 0
So, t = 0 or t = 90/4.9
Since we're looking for the positive time (ignoring the t = 0 solution), we find that t ≈ 18.4 seconds.
Now we can substitute this value of t back into the equation for horizontal motion:
s = (155.9)(t)
Substituting t = 18.4, we find:
s ≈ 2867 meters
So, the horizontal distance from the gun to the point where the projectile strikes the ground is approximately 2867 meters!