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 = 101 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, h (av) = 6.67 cm. What is the most accurate muzzle velocity that the soldier can report to his sergeant? _____ m/s
3
To find the most accurate muzzle velocity, we can use the equation of projectile motion:
h = (1/2) * g * t^2
Where:
h = vertical displacement (h(av) = 6.67 cm = 0.0667 m)
g = acceleration due to gravity (approximated as 9.8 m/s^2)
t = time of flight
In this case, since the bullet is fired horizontally, the time of flight (t) can be calculated using the horizontal distance (d) and the initial velocity (v0):
d = v0 * t
Solving for t:
t = d / v0
Now, substituting the value of t in the equation of projectile motion:
h = (1/2) * g * (d / v0)^2
Rearranging the equation to solve for v0:
v0 = sqrt((g * d^2) / (2 * h))
Substituting the given values:
v0 = sqrt((9.8 m/s^2 * (101 m)^2) / (2 * 0.0667 m))
Calculating the value:
v0 = sqrt(9919.8 m^2/s^2 / 0.1334 m)
v0 = sqrt(74348.0918 m^2/s^2)
v0 = 272.7 m/s
Therefore, the soldier can report the most accurate muzzle velocity as 272.7 m/s.
To find the muzzle velocity of the rifle, we can use the principles of projectile motion. In this case, since the bullet exits the barrel horizontally, the initial vertical velocity is zero.
Let's break down the problem and use the equation for projectile motion to find the muzzle velocity:
1. Start with the equation for vertical displacement in projectile motion:
h = (1/2) * g * t^2
Here, h is the vertical displacement, g is the acceleration due to gravity, and t is the time of flight.
2. Rewrite the equation to solve for time of flight, t:
t = sqrt((2h) / g)
Since the initial vertical velocity is zero, we can use this equation to find the time of flight.
3. Knowing the horizontal distance, d, and the time of flight, we can find the muzzle velocity using the equation:
d = v * t
Here, v is the muzzle velocity.
Let's calculate the muzzle velocity using the given values:
Given: d = 101 m, h(av) = 6.67 cm = 0.0667 m, g = 9.8 m/s^2
1. Calculate the time of flight, t:
t = sqrt((2 * h(av)) / g)
= sqrt((2 * 0.0667) / 9.8)
= 0.1157 s (rounded to four decimal places)
2. Calculate the muzzle velocity, v:
v = d / t
= 101 / 0.1157
= 872.20 m/s (rounded to two decimal places)
Therefore, the soldier can report the most accurate muzzle velocity of approximately 872.20 m/s to his sergeant.