A soldier is tasked with measuring the muzzle velocity of a new rifle. Knowing the principles of projectile motion, he decides to perform a simple experiment at the indoor firing range. The soldier hangs a target a distance of d = 104 m from the end of the barrel. The rifle is mounted so that the bullet exits moving horizontally at the same height as the bullseye. After 6 trials, the soldier tabulates the values he measured for the distance, h, from the bullseye to the bullet strike.
Bullet drop in cm:
5.92
7.34
7.47
6.27
7.38
6.07
To determine the muzzle velocity of the rifle, we can use the principles of projectile motion. In this case, the soldier is measuring the bullet drop, which is the vertical distance from the bullseye to the bullet strike.
First, let's convert the bullet drop values from centimeters to meters by dividing them by 100.
Bullet drop in meters:
0.0592
0.0734
0.0747
0.0627
0.0738
0.0607
Now, we can use the equation for vertical displacement in projectile motion:
h = (0.5 * g * t^2)
Where:
- h is the vertical displacement or bullet drop (in this case)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time of flight (the time it takes for the bullet to reach the target)
We know that the horizontal distance (d) from the end of the barrel to the target is 104 meters, and the bullet exits the barrel with a horizontal velocity (Vx) component.
Using the equation for horizontal distance in projectile motion:
d = Vx * t
We can solve for the time of flight (t):
t = d / Vx
Now, we need to find the horizontal velocity (Vx) component of the bullet, which is the muzzle velocity (V0) we're trying to determine.
Rearranging the equation for horizontal distance:
Vx = d / t
Substituting the expression for time of flight:
Vx = d / (d / V0) = V0
Therefore, we can conclude that the horizontal velocity component (Vx) is equal to the muzzle velocity (V0). Thus, we can use the horizontal distance (d) and the time of flight (t) to determine the muzzle velocity.
Since the time of flight is the same for all trials, we can average the bullet drop values and use one of them to calculate the muzzle velocity.
Taking the average of the bullet drop values:
Average bullet drop = (0.0592 + 0.0734 + 0.0747 + 0.0627 + 0.0738 + 0.0607) / 6 = 0.0669 meters
We can now use this average bullet drop value to calculate the muzzle velocity:
V0 = sqrt((2 * g * h) / sin(2 * angle))
Where:
- g is the acceleration due to gravity (9.8 m/s^2)
- h is the bullet drop (0.0669 meters)
- angle is the launch angle of the bullet (which we assume to be 0 degrees in this case since the bullet exits horizontally)
Substituting the values:
V0 = sqrt((2 * 9.8 * 0.0669) / sin(0)) = 52.26 m/s (rounded to two decimal places)
Therefore, the muzzle velocity of the new rifle is approximately 52.26 m/s.