The infamous German "Paris gun" was used to launch a projectile with a flight time of 170 s for a horizontal distance of 122 km. Based on this information, and ignoring air resistance , calculate (a) the gun's muzzle speed and (b) the angle, measured above the horizontal, at which it was fired.
Horizontal velocity component (a constant)
= Vx = 122,000/170 = 717.6 m/s
Vertical velocity component (initially)= Vy = g*85s = 833.0 m/s
Muzzle speed = sqrt[(717.6)^2 + (833)^2]
= 1100 m/s
Firing angle = tan^-1 (833/717.6) = 49.3 degrees
To calculate the muzzle speed and firing angle of the German "Paris gun," we can use the equations of motion for projectile motion. The key variables we need are the flight time and horizontal distance covered by the projectile.
(a) Calculating Muzzle Speed:
The muzzle speed (initial velocity) can be found using the equation:
v₀x = d / t
where v₀x is the initial horizontal velocity, d is the horizontal distance covered, and t is the flight time.
Substituting the given values:
v₀x = 122 km / 170 s
Note that we need to convert km to meters and seconds:
v₀x = (122 × 1000) m / 170 s
Simplifying:
v₀x = 720 m/s
So, the muzzle speed of the Paris gun is 720 m/s.
(b) Calculating Firing Angle:
To find the firing angle (θ) above the horizontal, we need to use the equation for horizontal distance:
d = v₀x × t
Rearranging the equation to find θ:
θ = tan⁻¹(d / v₀x)
Substituting the given values:
θ = tan⁻¹(122 km / 720 m/s)
Converting km to meters:
θ = tan⁻¹(122 × 1000 m / 720 m/s)
Simplifying:
θ = tan⁻¹(166667 m / 720 m/s)
Therefore, the firing angle (θ) is approximately equal to the arctangent of (166667 / 720):
θ ≈ 65.4°
So, the Paris gun was fired with an angle of approximately 65.4° above the horizontal.