A gun shoots bullets that leave the muzzle at 250 m/s. If a bullet is tohit a target 160.8 m away at the level of the muzzle, the gun must be aimedat a point above the target. (Neglect air resistance.)

Question

How far above the target is this point if the angle the gun makes is greater than 45°?

These are the tips they give me:
Use the formula for the range of a projectile.
I spent a whole day on this problem homies...HELP!!!

The range of the projectile is given by d = Vo^2(sin(2µ)/g where Vo = the initial muzzle velocity, g = the acceleration due to gravity and µ = the angle of the initial velocity vector aboc the horizontal. You have 3 of the pieces of information required. Substitute and solve for µ.

Answer 1

We have the range d = 160.8 m, the initial muzzle velocity Vo = 250 m/s, and we know that the angle µ is greater than 45°. The acceleration due to gravity g = 9.81 m/s².

We can start by solving for µ using the range formula:
d = Vo² * sin(2µ) / g
160.8 = (250)² * sin(2µ) / 9.81

Now we can solve for sin(2µ):
sin(2µ) = (160.8 * 9.81) / (250)²

Plugging in the values, we get:
sin(2µ) ≈ 0.4074

Now we can find the angle µ by finding the inverse sine (arcsin) of the value we got:
2µ = arcsin(0.4074)
2µ ≈ 24.56°

Now, we can find the angle µ by dividing by 2:
µ ≈ 12.28°

Since we know the angle should be greater than 45°, we should add 45° to this value:
µ > 45° + 12.28°
µ ≈ 57.28°

Now that we have the angle of the initial velocity vector above the horizontal, we can find the time it takes to travel the distance:
t = d / (Vo * cos(µ))

Plugging in the values:
t = 160.8 / (250 * cos(57.28°))
t ≈ 0.911 seconds

Next, we can find the upward motion of the projectile using the vertical component of the initial velocity vector and the time of flight:
Vertical motion = Vo * sin(µ) * t - (1/2) * g * t²

Plugging in the values:
Vertical motion = 250 * sin(57.28°) * 0.911 - (1/2) * 9.81 * (0.911)²

Calculating the vertical motion:
Vertical motion ≈ 111.1 m

So, the gun must be aimed approximately 111.1 meters above the target if the angle the gun makes is greater than 45°.