A gun that shoots bullets at 469m/s is to be aimed at a target 45.7 m away ans level with the gun. How high above the target must the gun barrel be pointed so that the bullet hits the target?

Which equation would I use?

I should have gotten 4.8 cm for the answer...

To solve this problem, you can use the equations of motion for projectile motion.

The equation you need to use is the vertical motion equation, which relates position, initial velocity, time, and acceleration. In this case, the acceleration is due to gravity and is equal to -9.8 m/s^2 (assuming that the motion is taking place near the Earth's surface and neglecting air resistance).

The equation for vertical displacement is given by:

Δy = Vyi * t + (1/2) * a * t^2

Where:
Δy is the vertical displacement (the height above the target in this case),
Vyi is the initial vertical velocity (the vertical component of the bullet's velocity),
t is the time of flight,

The initial vertical velocity, Vyi, can be found using trigonometry. Since the target is at the same level as the gun, the initial vertical velocity is equal to the product of the initial velocity (469 m/s) and the sine of the angle of elevation (θ).

Vyi = V0 * sin(θ)

In this case, we want to find the height above the target, so we set Δy equal to the distance to the target, which is 45.7 m. We also know that the time of flight is the same for both the vertical and horizontal motions.

Using these equations, we can solve for the angle of elevation (θ):

45.7 = (469 * sin(θ)) * t + (1/2) * (-9.8) * t^2

You would solve this equation for t using quadratic equation, and then substitute the value of t back into the equation to find the height above the target (Δy). Finally, convert the height to centimeters to match your expected result.

Keep in mind that the final outcome may not match your expected result exactly due to rounding errors or slight variations in the problem setup. It's important to double-check the calculations to ensure accuracy.