A rifle that shoots bullets at 447 m/s is to be aimed at a target 40.7 m away. If the center of the target is level with the rifle, how high above the target must the rifle barrel be pointed so that the bullet hits dead center?

To determine how high above the target the rifle barrel should be pointed, we can use the equation of motion for vertical motion:

y = y0 + v0y * t - (1/2) * g * t^2

Where:
y is the vertical displacement from the rifle barrel to the target
y0 is the initial vertical displacement, which is zero in this case (since the rifle and target are at the same level)
v0y is the vertical component of the initial velocity, which is the muzzle velocity of the bullet in this case
g is the acceleration due to gravity (-9.8 m/s^2)
t is the time it takes for the bullet to reach the target

First, let's calculate the time it takes for the bullet to travel the horizontal distance to the target. We can use the formula:

s = v0 * t

Where:
s is the horizontal distance to the target (40.7 m)
v0 is the initial velocity of the bullet (447 m/s)

t = s / v0

t = 40.7 m / 447 m/s
t ≈ 0.091s

Now, we can substitute this time into the equation of motion:

y = 0 + v0y * t - (1/2) * g * t^2

Since the initial vertical displacement (y0) is zero:

y = v0y * t - (1/2) * g * t^2

The vertical component of the initial velocity (v0y) can be calculated using trigonometry. We know that the muzzle velocity (v0) is 447 m/s. Assuming the angle between the rifle barrel and the horizontal is θ, we can use the sine function:

sin(θ) = v0y / v0

v0y = v0 * sin(θ)

To solve for θ, rearrange the equation:

sin(θ) = v0y / v0
sin(θ) = y / v0
θ = arcsin(y / v0)

Substituting the given values:

θ ≈ arcsin(40.7 m / 447 m/s)

Using a calculator, we find that θ ≈ 0.091 radians.

Finally, we can substitute the values into the equation of motion:

y = (447 m/s) * (0.091 s) - (1/2) * (9.8 m/s^2) * (0.091 s)^2

Now we can solve for the vertical displacement, y:

y ≈ (40.6 m) - (0.41 m)

y ≈ 40.2 m

Therefore, the rifle barrel must be pointed approximately 40.2 meters above the target to hit dead center.

To find out how high above the target the rifle barrel must be pointed, we need to determine the vertical displacement (height) of the bullet's trajectory.

We know that a bullet's motion can be treated as a projectile, following a parabolic path under the influence of gravity. The horizontal and vertical motions are independent of each other.

Let's break down the problem into two components: the horizontal motion and the vertical motion.

1. Horizontal Motion:
Since the bullet's initial velocity is purely horizontal, there is no acceleration in the horizontal direction. Therefore, the time it takes for the bullet to reach the target is determined solely by its horizontal displacement and the horizontal component of its velocity.
Using the formula:
time = distance / velocity

The time it takes for the bullet to reach the target is:
time = 40.7 m / 447 m/s

2. Vertical Motion:
In the vertical direction, we know that the bullet experiences constant acceleration due to gravity (-9.8 m/s^2). We need to calculate the time it takes for the bullet to reach the target in the vertical direction and find the corresponding displacement.

Using the formula for displacement in vertical motion:
displacement = initial velocity * time + (1/2) * acceleration * time^2

Since the initial vertical velocity is 0 m/s (the bullet starts at the same height as the target), the formula simplifies to:
displacement = (1/2) * acceleration * time^2

Substituting the values:
displacement = (1/2) * (-9.8 m/s^2) * (time)^2

Now, we can integrate the horizontal and vertical motions to find the total displacement or height above the target where the rifle barrel should be pointed.

Therefore, the calculation would involve substituting the value of time (found in the horizontal motion calculation) into the vertical motion displacement calculation to find the height.

Finally, subtract the height of the target itself (since it is at the same level as the rifle) to get the answer of how high above the target the rifle barrel must be pointed.