A rifle that shoots bullets at 460 m/s is to be aimed at a target 47.0 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? (Neglect air resistance.)
xo=x original vo=v original
So first in the horizontal direction
x=xo+vot+.5at^2
x=vot
t=47/460=0.102seconds to hit target
then for vertical direction
x=xo+vot+.5at^2
x=.5at^2
x=.5(9.8)(0.102)^2=0.0512 m above target
To determine how high above the target the rifle barrel must be pointed, we can use the equation of motion for vertical projectile motion. In this case, the projectile is the bullet.
Let's break down the problem and use the following variables:
Initial vertical position = 0 (level with the rifle barrel)
Final vertical position = ?
Initial vertical velocity = 0 (since the bullet is initially at the same height as the rifle)
Final vertical velocity = ?
Acceleration due to gravity, g = -9.8 m/s^2 (negative because it acts downwards)
Time of flight, t = ?
Using the equation of motion:
Final vertical position = Initial vertical position + (Initial vertical velocity * time) + (0.5 * acceleration due to gravity * time^2)
Since the initial vertical velocity is 0, this simplifies to:
Final vertical position = 0 + 0 + (0.5 * acceleration due to gravity * time^2)
Now, let's find the time of flight. The horizontal distance traveled is 47.0 m, and the horizontal velocity is 460 m/s. We can use the equation:
Horizontal distance = Horizontal velocity * time
47.0 m = 460 m/s * time
Solving for time, we have:
time = 47.0 m / 460 m/s
time ≈ 0.102 seconds
Now, let's substitute this value of time back into the equation for final vertical position:
Final vertical position = 0 + 0 + (0.5 * (-9.8 m/s^2) * (0.102 s)^2)
Final vertical position ≈ -0.050 m
To hit the dead center of the target, the rifle barrel must be pointed about 0.050 m (or 5.0 cm) above the target.
To find the height above the target the rifle barrel must be pointed, we need to take into account the horizontal and vertical components of the bullet's motion.
First, let's consider the horizontal component. The time it takes for the bullet to reach the target can be determined using the formula:
time = distance / velocity
In this case, the distance is 47.0 m and the velocity is 460 m/s:
time = 47.0 m / 460 m/s
time ≈ 0.102 sec
Now, let's focus on the vertical component. We can use the equation of motion to find the displacement of the bullet vertically:
displacement = initial velocity * time + (1/2) * acceleration * time^2
Since there is no vertical acceleration (neglecting air resistance), the equation simplifies to:
displacement = initial velocity * time
The initial velocity of the bullet in the vertical direction is 0 m/s because the bullet starts at the same height as the target. So:
displacement = 0 m/s * 0.102 sec
displacement = 0 m
This means that the bullet does not drop or rise during its flight.
Therefore, the rifle barrel must be pointed in line with the target without any vertical adjustment.