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. 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 ?
To find the greatest horizontal distance the cannonball can be shot, we need to determine the angle at which the cannon should be fired.
We know that the horizontal distance traveled by the cannonball is given by the equation:
d = (v^2 * sin(2θ))/g
Where:
- d is the horizontal distance
- v is the muzzle velocity (350 m/s)
- θ is the angle of elevation
- g is the acceleration due to gravity (9.8 m/s^2)
To maximize the horizontal distance, we need to find the angle that gives the maximum value for the expression (v^2 * sin(2θ))/g.
To find this angle, we can take the derivative of the expression with respect to θ and set it equal to 0 to find the maximum point.
d/dθ ((v^2 * sin(2θ))/g) = 0
2v^2 * cos(2θ) / g = 0
cos(2θ) = 0
2θ = π/2
θ = π/4
So the angle at which the cannon should be fired to achieve the maximum horizontal distance is θ = π/4 radians or 45 degrees.
Now, we can substitute the values into the equation to find the maximum horizontal distance:
d = (v^2 * sin(2θ))/g
= (350^2 * sin(2 * π/4))/9.8
= (122500 * 1)/9.8
≈ 12,500 m
Therefore, the greatest horizontal distance the cannonball can be shot is approximately 12,500 meters.