The cannon on a pirate ships shoots cannon balls with a speed of 350m/s(the muzzle velocity). The cannon can be adjusted to shoot at any elevation above the horizontal.

Question

Ignoring air resistance and assuming the cannon ball is shot from ground (or water) level, what is the greatest horizontal distance the cannon ball can be shot in m?

Answer 1

To find the greatest horizontal distance (range) that a cannonball can be shot from a cannon, we can use the formula for the range of a projectile launched at an angle \( \theta \) with an initial velocity \( v_0 \):

\[ R = \frac{v_0^2 \sin(2\theta)}{g} \]

Where:

• \( R \) is the range,
• \( v_0 \) is the initial velocity (350 m/s in this case),
• \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)),
• \( \theta \) is the launch angle.

To maximize the range, the angle \( \theta \) should be \( 45^\circ \). At this angle, \( \sin(2\theta) = \sin(90^\circ) = 1 \).

Substituting \( \theta = 45^\circ \):

\[ R = \frac{(350)^2 \cdot 1}{9.81} \]

Calculating \( 350^2 \):

\[ 350^2 = 122500 \]

Now substitute that back into the range formula:

\[ R = \frac{122500}{9.81} \approx 12482.24 , \text{m} \]

So, the greatest horizontal distance the cannonball can be shot is approximately \( \boxed{12482.24} , \text{m} \).