For this problem, ignore air resistance.

You are to fire a rifle that shoots a particular bullet at 460 m/s at a target that is 45.7 m away and level with the rifle. Assuming that the acceleration due to gravity is 9.80 m/s², how high above the target must the barrel of the rifle be aimed such that the bullet hits the target?

The real question is how far the bullet falls vertically during the trip.

timeinair=45.7m/460m/s

how far does an object fall in that time?

h=1/2 g t^2

That's not entirely correct... That would only be true if the rifle was fired horizontally, giving the x component 460 m/s towards the target.

However, that 460 m/s has some UPWARD y component too, reducing the x component by some amount...

Thanks,
Steven

Forget that upward component. Too fast to be significant. look at 460 cos (angle up)

the cosine of your angle up will be essentially 1 and the sine will be about 0

I would like to forget the upward component, as the object IS in fact a bullet fired at a fast velocity from a gun towards a target not too far away.

However, we're graded on the correctness of our problem solving. And, additionally, this factor WOULD become an issue if we were firing a marshmallow gun or a pea shooter at this distance, or if you were firing a sniper rifle at a target over a mile away.

Thanks,
Steven

To solve this problem, we need to use the kinematic equations of motion. We'll assume the initial vertical position of the rifle barrel is at the same height as the target, and we need to find the height the barrel must be aimed above the target to hit it.

Let's analyze the horizontal and vertical components of the motion separately.

1. For the horizontal motion:
- The horizontal distance traveled by the bullet is given as 45.7 m.
- The initial horizontal velocity (vx) of the bullet is equal to the horizontal component of the bullet's initial velocity, which is 460 m/s.
- Since there is no horizontal acceleration (air resistance is ignored), the horizontal component of velocity (vx) remains constant.
- Using the formula:
Distance = Velocity * Time,
we can find the time (t) taken for the bullet to travel the horizontal distance:
45.7 m = 460 m/s * t,
t = 45.7 m / 460 m/s,
t ≈ 0.099 s.

2. For the vertical motion:
- The initial vertical velocity (vy) of the bullet is zero (since the bullet is fired horizontally).
- The acceleration due to gravity (g) is -9.80 m/s² (negative because it acts downwards).
- We need to find the initial vertical displacement (y) of the bullet when it reaches the target (which will be the same as the height the barrel must be aimed above the target).
- Using the formula:
y = vy * t + (1/2) * g * t^2,
where vy = 0, t = 0.099 s, and g = -9.80 m/s²,
y = 0 * 0.099 + (1/2) * (-9.80) * (0.099)^2,
y = -0.0485 m.

Therefore, the barrel of the rifle must be aimed approximately 0.0485 meters (or 4.85 cm) above the target to hit it when ignoring air resistance.