A horizontal rifle is fired at a bull's-eye. The muzzle speed of the bullet is 640 m/s. The gun is pointed directly at the center of the bull's-eye, but the bullet strikes the target 0.028 m below the center. What is the horizontal distance between the end of the rifle and the bull's-eye?
To find the horizontal distance between the end of the rifle and the bull's-eye, we need to determine how long it takes for the bullet to reach the target and then multiply that time by the horizontal velocity of the bullet.
First, let's analyze the vertical motion of the bullet. The bullet is fired horizontally, so there is no initial vertical velocity. The only force acting on it is gravity, which causes it to accelerate downward.
We'll use the following equation to find the time of flight for the bullet:
y = v₀t + (1/2)gt²
Where:
y is the vertical displacement (-0.028 m, as the bullet strikes below the center),
v₀ is the initial vertical velocity (0 m/s since the bullet is fired horizontally),
t is the time of flight, and
g is the acceleration due to gravity (-9.8 m/s²).
Substituting the values into the equation, we get:
-0.028 = 0 + (1/2)(-9.8)t²
Simplifying the equation, we get:
-0.028 = -4.9t²
Dividing both sides by -4.9, we have:
t² = 0.028 / 4.9
Taking the square root of both sides, we get:
t = √(0.028 / 4.9)
Calculating this equation, we find that t is approximately 0.064 seconds.
Now that we have the time of flight, we can use the horizontal velocity of the bullet to find the horizontal distance traveled. The horizontal velocity remains constant throughout the flight because no horizontal forces act on the bullet.
The horizontal distance traveled can be found using the equation:
d = vxt
Where:
d is the horizontal distance,
vx is the horizontal velocity (which is the same as the muzzle speed, 640 m/s), and
t is the time of flight (0.064 seconds).
Substituting the values into the equation, we get:
d = 640 m/s × 0.064 s
Calculating this equation, we find that the horizontal distance traveled by the bullet is approximately 41.6 meters.
Therefore, the horizontal distance between the end of the rifle and the bull's-eye is approximately 41.6 meters.
To find the horizontal distance between the end of the rifle and the bull's-eye, we need to determine how far the bullet would have traveled horizontally before hitting the target.
Given:
Muzzle speed of the bullet (v) = 640 m/s
Vertical displacement of the bullet (h) = 0.028 m
First, let's calculate the time it takes for the bullet to reach the target. We can use the equation of motion for vertical displacement:
h = (1/2) * g * t^2
Where:
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time it takes for the bullet to reach the target
0.028 = (1/2) * 9.8 * t^2
0.056 = 4.9 * t^2
Now, let's solve for t:
t^2 = 0.056 / 4.9
t^2 = 0.01142
t ≈ √0.01142
t ≈ 0.107 s
Now that we have the time it takes for the bullet to reach the target, we can calculate the horizontal distance traveled by using the formula:
d = v * t
Where:
v is the muzzle speed of the bullet
t is the time taken to reach the target
d = 640 * 0.107
d ≈ 68.48 m
Therefore, the horizontal distance between the end of the rifle and the bull's-eye is approximately 68.48 meters.