The drawing shows an exaggerated view of a rifle that has been "sighted in" for a 91.4-meter target. If the muzzle speed of the bullet is vo=427m/s, what are the two possible angles between the rifle barrel and the horizontal such that the bullet will hit the target? One of these angles is so large that it is never used in target shooting.
To determine the two possible angles, we can use the concept of projectile motion. The key idea is that the motion of the bullet can be divided into horizontal and vertical components.
First, let's consider the vertical motion. The bullet will follow a parabolic trajectory due to the effect of gravity. At the highest point of its trajectory, the vertical velocity component becomes zero. We'll call this point the peak of the trajectory.
To find the angle of the barrel with the horizontal, we need to find the launch angle that will result in the bullet hitting the target 91.4 meters away.
1. Finding the time of flight:
The time it takes for the bullet to reach the target can be determined using the formula for horizontal motion:
distance = velocity × time
In this case, the distance is 91.4 meters, and the velocity is the horizontal component of the bullet's initial velocity, which remains constant throughout the motion. Therefore:
91.4 = 427 × time
Solving for time, we find:
time = 91.4 / 427
2. Finding the vertical displacement:
The vertical displacement of the bullet can be calculated using the formula:
displacement = initial velocity × time + (1/2) × acceleration × time^2
The initial vertical velocity is zero since the bullet is initially launched horizontally. Therefore, the formula simplifies to:
displacement = (1/2) × acceleration × time^2
For the peak of the trajectory, the vertical displacement will be equal to half the range, which is 91.4 / 2 meters. So:
(91.4 / 2) = (1/2) × (-9.8) × (time)^2
Simplifying this equation gives:
time = sqrt((91.4 / 2) / (-9.8))
3. Finding the launch angle:
Once we have the time of flight, we can find the launch angle by using the formula:
angle = atan(vertical velocity / horizontal velocity)
Since the vertical velocity is determined by the time of flight and the acceleration due to gravity, it can be calculated as:
vertical velocity = -9.8 × time
Now, to find the two possible angles, we substitute the horizontal velocity (vo = 427 m/s) and vertical velocity into the formula:
angle = atan((-9.8 × time) / 427)
By calculating this expression, we can find the launch angle for the two possible solutions. One of these angles will be reasonable for target shooting, while the other will be too large to be practical.