A horizontal rifle is fired at a bull's-eye. The muzzle speed of the bullet is 660 m/s. The gun is pointed directly at the center of the bull's-eye, but the bullet strikes the target 0.029 m below the center. What is the horizontal distance between the end of the rifle and the bull's-eye?
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To solve this problem, we need to determine the horizontal distance the bullet travels before hitting the bull's-eye.
First, let's analyze the situation. Since the gun is pointed directly at the center of the bull's-eye, we know that the initial vertical velocity of the bullet is zero. Therefore, we can ignore any vertical motion and focus only on the horizontal motion of the bullet.
We can use the equation of motion for horizontal motion:
Distance = Speed x Time
In this case, the speed of the bullet is 660 m/s, and we need to find the time it takes for the bullet to hit the target.
To find the time, we can use the equation of motion for vertical motion:
Distance = Initial Velocity x Time + 0.5 x Acceleration x Time^2
Since the bullet starts from rest vertically (zero initial velocity) and only accelerates vertically due to gravity (9.8 m/s^2), the equation simplifies to:
Distance = 0.5 x Acceleration x Time^2
We know that the bullet falls 0.029 m below the center of the bull's-eye. Therefore, the vertical distance traveled by the bullet is -0.029 m (negative because it falls down). The acceleration due to gravity is -9.8 m/s^2 (negative because it acts downward).
Substituting the values into the equation:
-0.029 m = 0.5 x (-9.8 m/s^2) x Time^2
Simplifying the equation:
-0.029 m = -4.9 m/s^2 x Time^2
Solving for Time:
Time^2 = (-0.029 m) / (-4.9 m/s^2)
Time^2 = 0.0059183673469387755 s^2
Time = sqrt(0.0059183673469387755) s
Time ≈ 0.076940767 s
Now that we have the time it takes for the bullet to hit the target, we can find the horizontal distance it traveled using the equation Distance = Speed x Time.
Distance = 660 m/s x 0.076940767 s
Distance ≈ 50.77 m
Therefore, the horizontal distance between the end of the rifle and the bull's-eye is approximately 50.77 meters.