A French artillerist, Captain A. Le Duc,

developed an empirical formula for the velocity of a
projectile in terms of the distance traveled by the
projectile in the bore. For a 155-mm gun this
equation is
v = 3309s /(70 + s)
where s is the distance traveled in the bore in inches
and v is the speed of the projectile in feet per second.
Given that the distance the projectile travels in the
bore is 185 in., determine: (a) the magnitude of the
muzzle velocity of the projectile; (b) the magnitude of
the acceleration of the projectile when s = 50 in.

Ah, Captain A. Le Duc, the Picasso of artillery equations! Let's calculate the answers to your questions with a touch of Clown Bot humor:

(a) To find the muzzle velocity of the projectile, let's plug in the value of s = 185 inches into the equation and see what pops out! *Drumroll, please!* According to my calculations, the muzzle velocity of the projectile is around 3151.645 feet per second. That's fast! Zoom, zoom!

(b) Now, to discover the acceleration of the projectile when s = 50 inches, we need to calculate the derivative of velocity with respect to distance. But fret not, my friend, Clown Bot is here to simplify things! Using Le Duc's equation, we find that the acceleration is approximately 270.1204 feet per second squared. Phew! That's quite a kick!

So, there you have it! Muzzle velocity around 3151.645 ft/s and acceleration around 270.1204 ft/s² when s = 50 inches. But remember, in the world of artillery, math can explode with excitement!

To determine the values requested, we will substitute the given values into the formula and perform the necessary calculations.

(a) To find the magnitude of the muzzle velocity of the projectile, we need to calculate the speed (v) when s = 185 inches.

Substituting s = 185 into the formula:
v = 3309(185) / (70 + 185)
Simplifying the equation:
v = 610065 / 255
v ≈ 2392.17 ft/s

Therefore, the magnitude of the muzzle velocity of the projectile is approximately 2392.17 ft/s.

(b) To find the magnitude of the acceleration of the projectile when s = 50 inches, we need to differentiate the formula with respect to s.

Differentiating the formula:
dv/ds = 3309(70 + s) - 3309s / (70 + s)^2

Substituting s = 50 into the derivative formula:
dv/ds = 3309(70 + 50) - 3309(50) / (70 + 50)^2
Simplifying the equation:
dv/ds = 3309(120) - 3309(50) / (120)^2
dv/ds = 397080 - 165450 / 14400
dv/ds ≈ 15.20 ft/s^2

Therefore, the magnitude of the acceleration of the projectile when s = 50 inches is approximately 15.20 ft/s^2.

To determine the answers to the given questions, we can substitute the given values into the formula and calculate the results. Let's solve each part separately:

(a) To find the magnitude of the muzzle velocity of the projectile, we need to determine the value of v when s = 185 inches.

Substituting s = 185 into the equation:
v = 3309(185) / (70 + 185)

Now, let's calculate it:
v = 611565 / 255
v ≈ 2398.29 ft/s

Therefore, the magnitude of the muzzle velocity of the projectile is approximately 2398.29 ft/s.

(b) To find the magnitude of the acceleration of the projectile when s = 50 inches, we need to determine the derivative of v with respect to s and evaluate it at s = 50.

Differentiating the equation with respect to s:
dv / ds = 3309 * (70 + s) - 3309s / (70 + s)^2

Substituting s = 50 into the derivative formula:
dv / ds = 3309 * (70 + 50) - 3309(50) / (70 + 50)^2

Now, let's calculate it:
dv / ds = 3309 * 120 - 3309(50) / 120^2
dv / ds = 397080 - 165450 / 14400
dv / ds ≈ 231630 / 14400
dv / ds ≈ 16.086 ft/s²

Therefore, the magnitude of the acceleration of the projectile when s = 50 inches is approximately 16.086 ft/s².