At a firing range, a target is placed 300 m from where someone is firing a rifle. The muzzle speed of this rifle is 700 m/s. If the rifle is aimed directly at the center of the target, by how far will the target miss? At what angle would the rifle need to be aimed at to hit the target dead center?
Range = Vo^2*sin(2A)/g = 300 m.
Vo = 700 m/s.
g = 9.8 m/s^2.
A = ?
To determine how far the target will be missed, we need to calculate the time it takes for the bullet to reach the target. Since the rifle is fired horizontally, the vertical component of the bullet's velocity is zero. Thus, we can treat this as a projectile motion problem, neglecting air resistance.
Let's start by calculating the time it takes for the bullet to reach the target.
Using the equation of motion for vertical displacement:
d = V₀t + (1/2)gt²
Where:
- d is the vertical displacement (which is zero in this case)
- V₀ is the initial vertical velocity (zero in this case)
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- t is the time
Since the vertical displacement is zero, the equation simplifies to:
0 = 0 + (1/2)gt²
Solving for t:
0 = 4.9t²
t = 0
This means that, after 0 seconds, the bullet reaches the target vertically.
Now, let's calculate the horizontal distance the bullet travels in that time.
Using the equation of motion for horizontal displacement:
d = V₀t
Where:
- d is the horizontal displacement
- V₀ is the initial horizontal velocity (equal to the muzzle speed of the rifle, 700 m/s)
- t is the time (0 seconds in this case)
d = 700 × 0
d = 0
Therefore, the bullet does not deviate horizontally and hits the target dead center.
To determine the angle at which the rifle should be aimed to hit the target dead center, we need to consider the vertical component of the bullet's motion.
Using the equation of motion for vertical displacement:
d = V₀sinθt - (1/2)gt²
Where:
- d is the vertical displacement (half the distance to the target, 150 m)
- V₀ is the initial velocity (700 m/s)
- θ is the angle of projection
- t is the time of flight
We know that the vertical displacement is half the distance to the target, so we can plug in the values:
150 = (700sinθ)t - 4.9t²
Since we already found that the time of flight is 0, the equation simplifies to:
150 = 0
This equation does not have a valid solution, which means there is no angle at which the rifle needs to be aimed to hit the target dead center. The rifle can be aimed directly at the center of the target without any adjustment.